diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index cf653a6875a9c..a1967c939092f 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -1,7 +1,7 @@
 """
 Base and utility classes for tseries type pandas objects.
 """
-from datetime import datetime
+from datetime import datetime, timedelta
 from typing import Any, List, Optional, Union, cast
 
 import numpy as np
@@ -17,14 +17,18 @@
     ensure_int64,
     ensure_platform_int,
     is_bool_dtype,
+    is_datetime64_any_dtype,
     is_dtype_equal,
     is_integer,
     is_list_like,
+    is_object_dtype,
     is_period_dtype,
     is_scalar,
+    is_timedelta64_dtype,
 )
 from pandas.core.dtypes.concat import concat_compat
 from pandas.core.dtypes.generic import ABCIndex, ABCIndexClass, ABCSeries
+from pandas.core.dtypes.missing import isna
 
 from pandas.core import algorithms
 from pandas.core.arrays import DatetimeArray, PeriodArray, TimedeltaArray
@@ -41,7 +45,8 @@
 from pandas.core.ops import get_op_result_name
 from pandas.core.tools.timedeltas import to_timedelta
 
-from pandas.tseries.frequencies import DateOffset
+from pandas.tseries.frequencies import DateOffset, to_offset
+from pandas.tseries.offsets import Tick
 
 _index_doc_kwargs = dict(ibase._index_doc_kwargs)
 
@@ -72,13 +77,33 @@ def wrapper(left, right):
     return wrapper
 
 
+def _make_wrapped_arith_op_with_freq(opname: str):
+    """
+    Dispatch the operation to the underlying ExtensionArray, and infer
+    the appropriate frequency for the result.
+    """
+    meth = make_wrapped_arith_op(opname)
+
+    def wrapped(self, other):
+        result = meth(self, other)
+        if result is NotImplemented:
+            return NotImplemented
+
+        new_freq = self._get_addsub_freq(other)
+        result._freq = new_freq
+        return result
+
+    wrapped.__name__ = opname
+    return wrapped
+
+
 @inherit_names(
     ["inferred_freq", "_isnan", "_resolution", "resolution"],
     DatetimeLikeArrayMixin,
     cache=True,
 )
 @inherit_names(
-    ["mean", "freq", "freqstr", "asi8", "_box_func"], DatetimeLikeArrayMixin,
+    ["mean", "asi8", "_box_func"], DatetimeLikeArrayMixin,
 )
 class DatetimeIndexOpsMixin(ExtensionIndex):
     """
@@ -446,10 +471,45 @@ def get_indexer_non_unique(self, target):
         return ensure_platform_int(indexer), missing
 
     # --------------------------------------------------------------------
+    # Arithmetic Methods
+
+    def _get_addsub_freq(self, other) -> Optional[DateOffset]:
+        """
+        Find the freq we expect the result of an addition/subtraction operation
+        to have.
+        """
+        if is_period_dtype(self.dtype):
+            # Only used for ops that stay PeriodDtype
+            return self.freq
+        elif self.freq is None:
+            return None
+        elif lib.is_scalar(other) and isna(other):
+            return None
+
+        elif isinstance(other, (Tick, timedelta, np.timedelta64)):
+            new_freq = None
+            if isinstance(self.freq, Tick):
+                new_freq = self.freq
+            return new_freq
+
+        elif isinstance(other, DateOffset):
+            # otherwise just DatetimeArray
+            return None  # TODO: Should we infer if it matches self.freq * n?
+        elif isinstance(other, (datetime, np.datetime64)):
+            return self.freq
+
+        elif is_timedelta64_dtype(other):
+            return None  # TODO: shouldnt we be able to do self.freq + other.freq?
+        elif is_object_dtype(other):
+            return None  # TODO: is this quite right?  sometimes we unpack singletons
+        elif is_datetime64_any_dtype(other):
+            return None  # TODO: shouldnt we be able to do self.freq + other.freq?
+        else:
+            raise NotImplementedError
 
-    __add__ = make_wrapped_arith_op("__add__")
+    __add__ = _make_wrapped_arith_op_with_freq("__add__")
+    __sub__ = _make_wrapped_arith_op_with_freq("__sub__")
     __radd__ = make_wrapped_arith_op("__radd__")
-    __sub__ = make_wrapped_arith_op("__sub__")
     __rsub__ = make_wrapped_arith_op("__rsub__")
     __pow__ = make_wrapped_arith_op("__pow__")
     __rpow__ = make_wrapped_arith_op("__rpow__")
@@ -558,7 +618,9 @@ def shift(self, periods=1, freq=None):
         Index.shift : Shift values of Index.
         PeriodIndex.shift : Shift values of PeriodIndex.
         """
-        result = self._data._time_shift(periods, freq=freq)
+        arr = self._data.view()
+        arr._freq = self.freq
+        result = arr._time_shift(periods, freq=freq)
         return type(self)(result, name=self.name)
 
     # --------------------------------------------------------------------
@@ -610,21 +672,40 @@ class DatetimeTimedeltaMixin(DatetimeIndexOpsMixin, Int64Index):
     _is_monotonic_increasing = Index.is_monotonic_increasing
     _is_monotonic_decreasing = Index.is_monotonic_decreasing
     _is_unique = Index.is_unique
+    _freq = lib.no_default
 
-    def _set_freq(self, freq):
+    @property
+    def freq(self):
+        """
+        In limited circumstances, our freq may differ from that of our _data.
         """
-        Set the _freq attribute on our underlying DatetimeArray.
+        if self._freq is not lib.no_default:
+            return self._freq
+        return self._data.freq
 
-        Parameters
-        ----------
-        freq : DateOffset, None, or "infer"
+    @property
+    def freqstr(self):
+        """
+        Return the frequency object as a string if its set, otherwise None.
         """
-        # GH#29843
-        self._data._with_freq(freq)
+        if self.freq is None:
+            return None
+        return self.freq.freqstr
 
     def _with_freq(self, freq):
         index = self.copy(deep=False)
-        index._set_freq(freq)
+        if freq is None:
+            # Even if we _can_ have a freq, we might want to set it to None
+            index._freq = None
+        elif len(self) == 0 and isinstance(freq, DateOffset):
+            # Always valid.  In the TimedeltaArray case, we assume this
+            #  is a Tick offset.
+            index._freq = freq
+        else:
+            assert freq == "infer", freq
+            freq = to_offset(self.inferred_freq)
+            index._freq = freq
+
         return index
 
     def _shallow_copy(self, values=None, name: Label = lib.no_default):
@@ -647,8 +728,7 @@ def _shallow_copy(self, values=None, name: Label = lib.no_default):
 
     @Appender(Index.difference.__doc__)
     def difference(self, other, sort=None):
-        new_idx = super().difference(other, sort=sort)
-        new_idx._set_freq(None)
+        new_idx = super().difference(other, sort=sort)._with_freq(None)
         return new_idx
 
     def intersection(self, other, sort=False):
@@ -693,7 +773,7 @@ def intersection(self, other, sort=False):
             result = Index.intersection(self, other, sort=sort)
             if isinstance(result, type(self)):
                 if result.freq is None:
-                    result._set_freq("infer")
+                    result = result._with_freq("infer")
             return result
 
         elif (
@@ -704,14 +784,7 @@ def intersection(self, other, sort=False):
             or (not self.is_monotonic or not other.is_monotonic)
         ):
             result = Index.intersection(self, other, sort=sort)
-
-            # Invalidate the freq of `result`, which may not be correct at
-            # this point, depending on the values.
-
-            result._set_freq(None)
-            result = self._shallow_copy(result._data, name=result.name)
-            if result.freq is None:
-                result._set_freq("infer")
+            result = result._with_freq("infer")
             return result
 
         # to make our life easier, "sort" the two ranges
@@ -781,10 +854,9 @@ def _fast_union(self, other, sort=None):
             left_start = left[0]
             loc = right.searchsorted(left_start, side="left")
             right_chunk = right._values[:loc]
-            dates = concat_compat([left._values, right_chunk])
-            result = self._shallow_copy(dates)
-            result._set_freq("infer")
+            dates = concat_compat((left._values, right_chunk))
             # TODO: can we infer that it has self.freq?
+            result = self._shallow_copy(dates)._with_freq("infer")
             return result
         else:
             left, right = other, self
@@ -797,9 +869,8 @@ def _fast_union(self, other, sort=None):
             loc = right.searchsorted(left_end, side="right")
             right_chunk = right._values[loc:]
             dates = concat_compat([left._values, right_chunk])
-            result = self._shallow_copy(dates)
-            result._set_freq("infer")
             # TODO: can we infer that it has self.freq?
+            result = self._shallow_copy(dates)._with_freq("infer")
             return result
         else:
             return left
@@ -816,7 +887,7 @@ def _union(self, other, sort):
         if this._can_fast_union(other):
             result = this._fast_union(other, sort=sort)
             if result.freq is None:
-                result._set_freq("infer")
+                result = result._with_freq("infer")
             return result
         else:
             i8self = Int64Index._simple_new(self.asi8, name=self.name)
diff --git a/pandas/core/indexes/period.py b/pandas/core/indexes/period.py
index 957c01c2dca96..d3c49c5ed0796 100644
--- a/pandas/core/indexes/period.py
+++ b/pandas/core/indexes/period.py
@@ -71,7 +71,7 @@ def _new_PeriodIndex(cls, **d):
     PeriodArray,
     wrap=True,
 )
-@inherit_names(["is_leap_year", "freq", "_format_native_types"], PeriodArray)
+@inherit_names(["is_leap_year", "freq", "freqstr", "_format_native_types"], PeriodArray)
 class PeriodIndex(DatetimeIndexOpsMixin, Int64Index):
     """
     Immutable ndarray holding ordinal values indicating regular periods in time.
diff --git a/pandas/core/resample.py b/pandas/core/resample.py
index bfef4f63e2e8a..06751d9c35fab 100644
--- a/pandas/core/resample.py
+++ b/pandas/core/resample.py
@@ -1017,7 +1017,8 @@ def _downsample(self, how, **kwargs):
         if not len(ax):
             # reset to the new freq
             obj = obj.copy()
-            obj.index._set_freq(self.freq)
+            obj.index = obj.index._with_freq(self.freq)
+            assert obj.index.freq == self.freq, (obj.index.freq, self.freq)
             return obj
 
         # do we have a regular frequency
diff --git a/pandas/tests/arithmetic/test_datetime64.py b/pandas/tests/arithmetic/test_datetime64.py
index 79fcb5e9478c3..912ce2a953e0a 100644
--- a/pandas/tests/arithmetic/test_datetime64.py
+++ b/pandas/tests/arithmetic/test_datetime64.py
@@ -2052,7 +2052,7 @@ def test_dti_add_tdi(self, tz_naive_fixture):
         dti = DatetimeIndex([Timestamp("2017-01-01", tz=tz)] * 10)
         tdi = pd.timedelta_range("0 days", periods=10)
         expected = pd.date_range("2017-01-01", periods=10, tz=tz)
-        expected._set_freq(None)
+        expected = expected._with_freq(None)
 
         # add with TimdeltaIndex
         result = dti + tdi
@@ -2074,7 +2074,7 @@ def test_dti_iadd_tdi(self, tz_naive_fixture):
         dti = DatetimeIndex([Timestamp("2017-01-01", tz=tz)] * 10)
         tdi = pd.timedelta_range("0 days", periods=10)
         expected = pd.date_range("2017-01-01", periods=10, tz=tz)
-        expected._set_freq(None)
+        expected = expected._with_freq(None)
 
         # iadd with TimdeltaIndex
         result = DatetimeIndex([Timestamp("2017-01-01", tz=tz)] * 10)
diff --git a/pandas/tests/arithmetic/test_timedelta64.py b/pandas/tests/arithmetic/test_timedelta64.py
index 3ffdc87ff84c8..9378a70044d83 100644
--- a/pandas/tests/arithmetic/test_timedelta64.py
+++ b/pandas/tests/arithmetic/test_timedelta64.py
@@ -544,7 +544,7 @@ def test_tda_add_sub_index(self):
     def test_tda_add_dt64_object_array(self, box_df_fail, tz_naive_fixture):
         # Result should be cast back to DatetimeArray
         dti = pd.date_range("2016-01-01", periods=3, tz=tz_naive_fixture)
-        dti._set_freq(None)
+        dti = dti._with_freq(None)
         tdi = dti - dti
 
         obj = tm.box_expected(tdi, box_df_fail)
diff --git a/pandas/tests/indexes/common.py b/pandas/tests/indexes/common.py
index 957ca138498d9..52b82b36d13be 100644
--- a/pandas/tests/indexes/common.py
+++ b/pandas/tests/indexes/common.py
@@ -267,7 +267,7 @@ def test_ensure_copied_data(self, indices):
         if is_datetime64tz_dtype(indices.dtype):
             result = result.tz_localize("UTC").tz_convert(indices.tz)
         if isinstance(indices, (DatetimeIndex, TimedeltaIndex)):
-            indices._set_freq(None)
+            indices = indices._with_freq(None)
 
         tm.assert_index_equal(indices, result)
 
@@ -397,7 +397,7 @@ def test_where(self, klass):
         i = self.create_index()
         if isinstance(i, (pd.DatetimeIndex, pd.TimedeltaIndex)):
             # where does not preserve freq
-            i._set_freq(None)
+            i = i._with_freq(None)
 
         cond = [True] * len(i)
         result = i.where(klass(cond))
diff --git a/pandas/tests/indexes/datetimelike.py b/pandas/tests/indexes/datetimelike.py
index 944358b1540b0..dfefdc0f211b1 100644
--- a/pandas/tests/indexes/datetimelike.py
+++ b/pandas/tests/indexes/datetimelike.py
@@ -82,7 +82,7 @@ def test_map_dictlike(self, mapper):
 
         # don't compare the freqs
         if isinstance(expected, (pd.DatetimeIndex, pd.TimedeltaIndex)):
-            expected._set_freq(None)
+            expected = expected._with_freq(None)
 
         result = index.map(mapper(expected, index))
         tm.assert_index_equal(result, expected)
diff --git a/pandas/tests/indexes/datetimes/test_constructors.py b/pandas/tests/indexes/datetimes/test_constructors.py
index a8e08bbe9a2e9..691f542fc2084 100644
--- a/pandas/tests/indexes/datetimes/test_constructors.py
+++ b/pandas/tests/indexes/datetimes/test_constructors.py
@@ -131,7 +131,7 @@ def test_construction_with_alt(self, kwargs, tz_aware_fixture):
     def test_construction_with_alt_tz_localize(self, kwargs, tz_aware_fixture):
         tz = tz_aware_fixture
         i = pd.date_range("20130101", periods=5, freq="H", tz=tz)
-        i._set_freq(None)
+        i = i._with_freq(None)
         kwargs = {key: attrgetter(val)(i) for key, val in kwargs.items()}
 
         if "tz" in kwargs:
diff --git a/pandas/tests/indexes/datetimes/test_datetime.py b/pandas/tests/indexes/datetimes/test_datetime.py
index 08706dce7e1e0..81fa1a27ac911 100644
--- a/pandas/tests/indexes/datetimes/test_datetime.py
+++ b/pandas/tests/indexes/datetimes/test_datetime.py
@@ -463,6 +463,5 @@ def test_split_non_utc(self):
         # GH 14042
         indices = pd.date_range("2016-01-01 00:00:00+0200", freq="S", periods=10)
         result = np.split(indices, indices_or_sections=[])[0]
-        expected = indices.copy()
-        expected._set_freq(None)
+        expected = indices._with_freq(None)
         tm.assert_index_equal(result, expected)
diff --git a/pandas/tests/indexes/datetimes/test_ops.py b/pandas/tests/indexes/datetimes/test_ops.py
index f0fe5e9b293fc..603a0a452391c 100644
--- a/pandas/tests/indexes/datetimes/test_ops.py
+++ b/pandas/tests/indexes/datetimes/test_ops.py
@@ -134,7 +134,7 @@ def test_value_counts_unique(self, tz_naive_fixture):
 
         exp_idx = pd.date_range("2011-01-01 18:00", freq="-1H", periods=10, tz=tz)
         expected = Series(range(10, 0, -1), index=exp_idx, dtype="int64")
-        expected.index._set_freq(None)
+        expected.index = expected.index._with_freq(None)
 
         for obj in [idx, Series(idx)]:
 
@@ -406,6 +406,20 @@ def test_freq_setter_errors(self):
         with pytest.raises(ValueError, match="Invalid frequency"):
             idx._data.freq = "foo"
 
+    def test_freq_view_safe(self):
+        # Setting the freq for one DatetimeIndex shouldn't alter the freq
+        #  for another that views the same data
+
+        dti = pd.date_range("2016-01-01", periods=5)
+        dta = dti._data
+
+        dti2 = DatetimeIndex(dta)._with_freq(None)
+        assert dti2.freq is None
+
+        # Original was not altered
+        assert dti.freq == "D"
+        assert dta.freq == "D"
+
 
 class TestBusinessDatetimeIndex:
     def setup_method(self, method):
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index df6e2dac72f95..0473ecf9de24d 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -231,9 +231,7 @@ def test_intersection(self, tz, sort):
         ]:
             result = base.intersection(rng)
             tm.assert_index_equal(result, expected)
-            assert result.name == expected.name
             assert result.freq == expected.freq
-            assert result.tz == expected.tz
 
         # non-monotonic
         base = DatetimeIndex(
@@ -255,6 +253,7 @@ def test_intersection(self, tz, sort):
         # GH 7880
         rng4 = date_range("7/1/2000", "7/31/2000", freq="D", tz=tz, name="idx")
         expected4 = DatetimeIndex([], tz=tz, name="idx")
+        assert expected4.freq is None
 
         for (rng, expected) in [
             (rng2, expected2),
@@ -265,9 +264,7 @@ def test_intersection(self, tz, sort):
             if sort is None:
                 expected = expected.sort_values()
             tm.assert_index_equal(result, expected)
-            assert result.name == expected.name
             assert result.freq is None
-            assert result.tz == expected.tz
 
     # parametrize over both anchored and non-anchored freqs, as they
     #  have different code paths
diff --git a/pandas/tests/indexes/timedeltas/test_ops.py b/pandas/tests/indexes/timedeltas/test_ops.py
index 0e5abe2f5ccd1..19cbd74b31172 100644
--- a/pandas/tests/indexes/timedeltas/test_ops.py
+++ b/pandas/tests/indexes/timedeltas/test_ops.py
@@ -290,3 +290,17 @@ def test_freq_setter_errors(self):
         # setting with non-freq string
         with pytest.raises(ValueError, match="Invalid frequency"):
             idx._data.freq = "foo"
+
+    def test_freq_view_safe(self):
+        # Setting the freq for one TimedeltaIndex shouldn't alter the freq
+        #  for another that views the same data
+
+        tdi = TimedeltaIndex(["0 days", "2 days", "4 days"], freq="2D")
+        tda = tdi._data
+
+        tdi2 = TimedeltaIndex(tda)._with_freq(None)
+        assert tdi2.freq is None
+
+        # Original was not altered
+        assert tdi.freq == "2D"
+        assert tda.freq == "2D"
diff --git a/pandas/tests/indexes/timedeltas/test_timedelta.py b/pandas/tests/indexes/timedeltas/test_timedelta.py
index f724badd51da8..637a2629dda8a 100644
--- a/pandas/tests/indexes/timedeltas/test_timedelta.py
+++ b/pandas/tests/indexes/timedeltas/test_timedelta.py
@@ -32,7 +32,9 @@ def indices(self):
     def create_index(self) -> TimedeltaIndex:
         index = pd.to_timedelta(range(5), unit="d")._with_freq("infer")
         assert index.freq == "D"
-        return index + pd.offsets.Hour(1)
+        ret = index + pd.offsets.Hour(1)
+        assert ret.freq == "D"
+        return ret
 
     def test_numeric_compat(self):
         # Dummy method to override super's version; this test is now done
diff --git a/pandas/tests/io/test_parquet.py b/pandas/tests/io/test_parquet.py
index e70a06cc5f582..280424c68297f 100644
--- a/pandas/tests/io/test_parquet.py
+++ b/pandas/tests/io/test_parquet.py
@@ -386,7 +386,7 @@ def test_write_index(self, engine):
         for index in indexes:
             df.index = index
             if isinstance(index, pd.DatetimeIndex):
-                index._set_freq(None)  # freq doesnt round-trip
+                df.index = df.index._with_freq(None)  # freq doesnt round-trip
             check_round_trip(df, engine, check_names=check_names)
 
         # index with meta-data
@@ -465,7 +465,7 @@ def test_basic(self, pa, df_full):
 
         # additional supported types for pyarrow
         dti = pd.date_range("20130101", periods=3, tz="Europe/Brussels")
-        dti._set_freq(None)  # freq doesnt round-trip
+        dti = dti._with_freq(None)  # freq doesnt round-trip
         df["datetime_tz"] = dti
         df["bool_with_none"] = [True, None, True]
 
@@ -634,7 +634,7 @@ def test_basic(self, fp, df_full):
         df = df_full
 
         dti = pd.date_range("20130101", periods=3, tz="US/Eastern")
-        dti._set_freq(None)  # freq doesnt round-trip
+        dti = dti._with_freq(None)  # freq doesnt round-trip
         df["datetime_tz"] = dti
         df["timedelta"] = pd.timedelta_range("1 day", periods=3)
         check_round_trip(df, fp)
diff --git a/pandas/tests/series/test_arithmetic.py b/pandas/tests/series/test_arithmetic.py
index 16163ee76ba63..c7a04843b8296 100644
--- a/pandas/tests/series/test_arithmetic.py
+++ b/pandas/tests/series/test_arithmetic.py
@@ -378,7 +378,7 @@ def test_ser_cmp_result_names(self, names, op):
 
         # datetime64tz dtype
         dti = dti.tz_localize("US/Central")
-        dti._set_freq("infer")  # freq not preserved by tz_localize
+        dti = pd.DatetimeIndex(dti, freq="infer")  # freq not preserved by tz_localize
         ser = Series(dti).rename(names[1])
         result = op(ser, dti)
         assert result.name == names[2]
diff --git a/pandas/tests/test_multilevel.py b/pandas/tests/test_multilevel.py
index 43461d465b9e7..1ba73292dc0b4 100644
--- a/pandas/tests/test_multilevel.py
+++ b/pandas/tests/test_multilevel.py
@@ -1505,7 +1505,7 @@ def test_set_index_datetime(self):
             tz="US/Eastern",
         )
         idx3 = pd.date_range("2011-01-01 09:00", periods=6, tz="Asia/Tokyo")
-        idx3._set_freq(None)
+        idx3 = idx3._with_freq(None)
 
         df = df.set_index(idx1)
         df = df.set_index(idx2, append=True)