diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 40adeb28d47b6..2c8ad1782af4d 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -547,6 +547,7 @@ Datetimelike
 - Bug in :meth:`DatetimeIndex.intersection` losing ``freq`` and timezone in some cases (:issue:`33604`)
 - Bug in :class:`DatetimeIndex` addition and subtraction with some types of :class:`DateOffset` objects incorrectly retaining an invalid ``freq`` attribute (:issue:`33779`)
 - Bug in :class:`DatetimeIndex` where setting the ``freq`` attribute on an index could silently change the ``freq`` attribute on another index viewing the same data (:issue:`33552`)
+- Bug in :meth:`DatetimeIndex.intersection` and :meth:`TimedeltaIndex.intersection` with results not having the correct ``name`` attribute (:issue:`33904`)
 
 Timedelta
 ^^^^^^^^^
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index fe48f05820a4a..3f18f69149658 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -6,7 +6,7 @@
 
 import numpy as np
 
-from pandas._libs import NaT, iNaT, join as libjoin, lib
+from pandas._libs import NaT, Timedelta, iNaT, join as libjoin, lib
 from pandas._libs.tslibs import timezones
 from pandas._typing import Label
 from pandas.compat.numpy import function as nv
@@ -41,7 +41,7 @@
 from pandas.core.sorting import ensure_key_mapped
 from pandas.core.tools.timedeltas import to_timedelta
 
-from pandas.tseries.frequencies import DateOffset
+from pandas.tseries.offsets import DateOffset, Tick
 
 _index_doc_kwargs = dict(ibase._index_doc_kwargs)
 
@@ -641,6 +641,7 @@ def intersection(self, other, sort=False):
         """
         self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
+        res_name = get_op_result_name(self, other)
 
         if self.equals(other):
             return self._get_reconciled_name_object(other)
@@ -654,12 +655,18 @@ def intersection(self, other, sort=False):
             result = Index.intersection(self, other, sort=sort)
             if isinstance(result, type(self)):
                 if result.freq is None:
+                    # TODO: no tests rely on this; needed?
                     result = result._with_freq("infer")
+            assert result.name == res_name
             return result
 
         elif not self._can_fast_intersect(other):
             result = Index.intersection(self, other, sort=sort)
-            result = result._with_freq("infer")
+            assert result.name == res_name
+            # We need to invalidate the freq because Index.intersection
+            #  uses _shallow_copy on a view of self._data, which will preserve
+            #  self.freq if we're not careful.
+            result = result._with_freq(None)._with_freq("infer")
             return result
 
         # to make our life easier, "sort" the two ranges
@@ -674,27 +681,34 @@ def intersection(self, other, sort=False):
         start = right[0]
 
         if end < start:
-            return type(self)(data=[], dtype=self.dtype, freq=self.freq)
+            return type(self)(data=[], dtype=self.dtype, freq=self.freq, name=res_name)
         else:
             lslice = slice(*left.slice_locs(start, end))
             left_chunk = left._values[lslice]
-            return self._shallow_copy(left_chunk)
+            return type(self)._simple_new(left_chunk, name=res_name)
 
     def _can_fast_intersect(self: _T, other: _T) -> bool:
         if self.freq is None:
             return False
 
-        if other.freq != self.freq:
+        elif other.freq != self.freq:
             return False
 
-        if not self.is_monotonic_increasing:
+        elif not self.is_monotonic_increasing:
             # Because freq is not None, we must then be monotonic decreasing
             return False
 
-        if not self.freq.is_anchored():
-            # If freq is not anchored, then despite having matching freqs,
-            #  we might not "line up"
-            return False
+        elif self.freq.is_anchored():
+            # this along with matching freqs ensure that we "line up",
+            #  so intersection will preserve freq
+            return True
+
+        elif isinstance(self.freq, Tick):
+            # We "line up" if and only if the difference between two of our points
+            #  is a multiple of our freq
+            diff = self[0] - other[0]
+            remainder = diff % self.freq.delta
+            return remainder == Timedelta(0)
 
         return True
 
@@ -749,6 +763,7 @@ def _fast_union(self, other, sort=None):
             right_chunk = right._values[:loc]
             dates = concat_compat((left._values, right_chunk))
             # With sort being False, we can't infer that result.freq == self.freq
+            # TODO: no tests rely on the _with_freq("infer"); needed?
             result = self._shallow_copy(dates)._with_freq("infer")
             return result
         else:
@@ -781,9 +796,12 @@ def _union(self, other, sort):
 
         if this._can_fast_union(other):
             result = this._fast_union(other, sort=sort)
-            if result.freq is None:
+            if sort is None:
                 # In the case where sort is None, _can_fast_union
                 #  implies that result.freq should match self.freq
+                assert result.freq == self.freq, (result.freq, self.freq)
+            elif result.freq is None:
+                # TODO: no tests rely on this; needed?
                 result = result._with_freq("infer")
             return result
         else:
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index 0473ecf9de24d..6670b079ddd29 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -222,7 +222,7 @@ def test_intersection(self, tz, sort):
         expected3 = date_range("6/1/2000", "6/20/2000", freq="D", name=None)
 
         rng4 = date_range("7/1/2000", "7/31/2000", freq="D", name="idx")
-        expected4 = DatetimeIndex([], name="idx")
+        expected4 = DatetimeIndex([], freq="D", name="idx")
 
         for (rng, expected) in [
             (rng2, expected2),
@@ -264,7 +264,7 @@ def test_intersection(self, tz, sort):
             if sort is None:
                 expected = expected.sort_values()
             tm.assert_index_equal(result, expected)
-            assert result.freq is None
+            assert result.freq == expected.freq
 
     # parametrize over both anchored and non-anchored freqs, as they
     #  have different code paths
diff --git a/pandas/tests/indexes/timedeltas/test_setops.py b/pandas/tests/indexes/timedeltas/test_setops.py
index d7576697435a0..6a2f66cade733 100644
--- a/pandas/tests/indexes/timedeltas/test_setops.py
+++ b/pandas/tests/indexes/timedeltas/test_setops.py
@@ -156,7 +156,7 @@ def test_zero_length_input_index(self, sort):
             # if no overlap exists return empty index
             (
                 timedelta_range("1 day", periods=10, freq="h", name="idx")[5:],
-                TimedeltaIndex([], name="idx"),
+                TimedeltaIndex([], freq="h", name="idx"),
             ),
         ],
     )
diff --git a/pandas/tests/indexing/test_partial.py b/pandas/tests/indexing/test_partial.py
index ee87a3f9f5ecd..a1bd6fed32cad 100644
--- a/pandas/tests/indexing/test_partial.py
+++ b/pandas/tests/indexing/test_partial.py
@@ -128,7 +128,7 @@ def test_partial_setting(self):
         df.at[dates[-1] + dates.freq, "A"] = 7
         tm.assert_frame_equal(df, expected)
 
-        exp_other = DataFrame({0: 7}, index=[dates[-1] + dates.freq])
+        exp_other = DataFrame({0: 7}, index=dates[-1:] + dates.freq)
         expected = pd.concat([df_orig, exp_other], axis=1)
 
         df = df_orig.copy()