diff --git a/pandas/_libs/tslibs/c_timestamp.pxd b/pandas/_libs/tslibs/c_timestamp.pxd
index d095b6027d2f9..49e7992063aa0 100644
--- a/pandas/_libs/tslibs/c_timestamp.pxd
+++ b/pandas/_libs/tslibs/c_timestamp.pxd
@@ -11,7 +11,7 @@ cdef class _Timestamp(datetime):
     cpdef _get_date_name_field(self, object field, object locale)
     cdef int64_t _maybe_convert_value_to_local(self)
     cpdef to_datetime64(self)
-    cdef _assert_tzawareness_compat(_Timestamp self, datetime other)
+    cdef _assert_tzawareness_compat(_Timestamp self, datetime other, int op)
     cpdef datetime to_pydatetime(_Timestamp self, bint warn=*)
     cdef bint _compare_outside_nanorange(_Timestamp self, datetime other,
                                          int op) except -1
diff --git a/pandas/_libs/tslibs/c_timestamp.pyx b/pandas/_libs/tslibs/c_timestamp.pyx
index 68987030e8b4e..079d8c49a55fb 100644
--- a/pandas/_libs/tslibs/c_timestamp.pyx
+++ b/pandas/_libs/tslibs/c_timestamp.pyx
@@ -131,7 +131,8 @@ cdef class _Timestamp(datetime):
             else:
                 return NotImplemented
 
-        self._assert_tzawareness_compat(other)
+        if self._assert_tzawareness_compat(other, op) is False:
+            return op == Py_NE
         return cmp_scalar(self.value, ots.value, op)
 
     def __reduce_ex__(self, protocol):
@@ -170,7 +171,8 @@ cdef class _Timestamp(datetime):
         cdef:
             datetime dtval = self.to_pydatetime()
 
-        self._assert_tzawareness_compat(other)
+        if self._assert_tzawareness_compat(other, op) is False:
+            return op == Py_NE
 
         if self.nanosecond == 0:
             return PyObject_RichCompareBool(dtval, other, op)
@@ -188,12 +190,18 @@ cdef class _Timestamp(datetime):
             elif op == Py_GE:
                 return dtval >= other
 
-    cdef _assert_tzawareness_compat(_Timestamp self, datetime other):
+    cdef _assert_tzawareness_compat(_Timestamp self, datetime other, int op):
+        is_inequality = not (op == Py_EQ or op == Py_NE)
+
         if self.tzinfo is None:
             if other.tzinfo is not None:
+                if not is_inequality:
+                    return False
                 raise TypeError('Cannot compare tz-naive and tz-aware '
                                 'timestamps')
         elif other.tzinfo is None:
+            if not is_inequality:
+                return False
             raise TypeError('Cannot compare tz-naive and tz-aware timestamps')
 
     cpdef datetime to_pydatetime(_Timestamp self, bint warn=True):
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index 30a34282889f8..595ce31cb7449 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -73,7 +73,10 @@ def wrapper(self, other):
 
         if isinstance(other, self._recognized_scalars) or other is NaT:
             other = self._scalar_type(other)
-            self._check_compatible_with(other)
+            if self._check_compatible_with(other, op=op) is False:
+                if op is operator.ne:
+                    return np.ones(self.shape, dtype=bool)
+                return np.zeros(self.shape, dtype=bool)
 
             other_i8 = self._unbox_scalar(other)
 
@@ -111,7 +114,10 @@ def wrapper(self, other):
             else:
                 # For PeriodDType this casting is unnecessary
                 other = type(self)._from_sequence(other)
-                self._check_compatible_with(other)
+                if self._check_compatible_with(other, op=op) is False:
+                    if op is operator.eq:
+                        return np.zeros(self.shape, dtype=bool)
+                    return np.ones(self.shape, dtype=bool)
 
                 result = op(self.view("i8"), other.view("i8"))
                 o_mask = other._isnan
@@ -187,7 +193,10 @@ def _unbox_scalar(self, value: Union[Period, Timestamp, Timedelta, NaTType]) ->
         raise AbstractMethodError(self)
 
     def _check_compatible_with(
-        self, other: Union[Period, Timestamp, Timedelta, NaTType], setitem: bool = False
+        self,
+        other: Union[Period, Timestamp, Timedelta, NaTType],
+        setitem: bool = False,
+        op=None,
     ) -> None:
         """
         Verify that `self` and `other` are compatible.
@@ -204,6 +213,7 @@ def _check_compatible_with(
         setitem : bool, default False
             For __setitem__ we may have stricter compatibility resrictions than
             for comparisons.
+        op : None or an operator.{eq,ne,lt,le,gt,ge} function
 
         Raises
         ------
@@ -853,7 +863,7 @@ def searchsorted(self, value, side="left", sorter=None):
             raise TypeError(f"Unexpected type for 'value': {type(value)}")
 
         if isinstance(value, type(self)):
-            self._check_compatible_with(value)
+            self._check_compatible_with(value)  # TODO: this is operator.lt?
             value = value.asi8
         else:
             value = self._unbox_scalar(value)
diff --git a/pandas/core/arrays/datetimes.py b/pandas/core/arrays/datetimes.py
index b9f9edcebad5b..4a5e4c5830637 100644
--- a/pandas/core/arrays/datetimes.py
+++ b/pandas/core/arrays/datetimes.py
@@ -1,4 +1,5 @@
 from datetime import datetime, time, timedelta
+import operator
 from typing import Union
 import warnings
 
@@ -455,10 +456,11 @@ def _unbox_scalar(self, value):
     def _scalar_from_string(self, value):
         return Timestamp(value, tz=self.tz)
 
-    def _check_compatible_with(self, other, setitem: bool = False):
+    def _check_compatible_with(self, other, setitem: bool = False, op=None):
         if other is NaT:
             return
-        self._assert_tzawareness_compat(other)
+        if self._assert_tzawareness_compat(other, op=op) is False:
+            return False
         if setitem:
             # Stricter check for setitem vs comparison methods
             if not timezones.tz_compare(self.tz, other.tz):
@@ -627,10 +629,15 @@ def _has_same_tz(self, other):
             # convert to Timestamp as np.datetime64 doesn't have tz attr
             other = Timestamp(other)
         vzone = timezones.get_timezone(getattr(other, "tzinfo", "__no_tz__"))
-        return zzone == vzone
+        return zzone == vzone  # TODO: use tz_compare?
 
-    def _assert_tzawareness_compat(self, other):
+    def _assert_tzawareness_compat(self, other, op=None):
         # adapted from _Timestamp._assert_tzawareness_compat
+
+        # GH#???? match stdlib behavior
+        is_inequality = not (op is operator.eq or op is operator.ne)
+
+        msg = "Cannot compare tz-naive and tz-aware datetime-like objects."
         other_tz = getattr(other, "tzinfo", None)
         if is_datetime64tz_dtype(other):
             # Get tzinfo from Series dtype
@@ -638,15 +645,12 @@ def _assert_tzawareness_compat(self, other):
         if other is NaT:
             # pd.NaT quacks both aware and naive
             pass
-        elif self.tz is None:
-            if other_tz is not None:
-                raise TypeError(
-                    "Cannot compare tz-naive and tz-aware datetime-like objects."
-                )
-        elif other_tz is None:
-            raise TypeError(
-                "Cannot compare tz-naive and tz-aware datetime-like objects"
-            )
+        elif int(self.tz is None) + int(other_tz is None) == 1:
+            # i.e. one but not the other
+            if is_inequality:
+                raise TypeError(msg)
+            return False
+        return None
 
     # -----------------------------------------------------------------
     # Arithmetic Methods
diff --git a/pandas/core/arrays/period.py b/pandas/core/arrays/period.py
index 99d9d69d66ec2..0f0f2cc840bf4 100644
--- a/pandas/core/arrays/period.py
+++ b/pandas/core/arrays/period.py
@@ -258,10 +258,10 @@ def _unbox_scalar(self, value: Union[Period, NaTType]) -> int:
     def _scalar_from_string(self, value: str) -> Period:
         return Period(value, freq=self.freq)
 
-    def _check_compatible_with(self, other, setitem: bool = False):
+    def _check_compatible_with(self, other, setitem: bool = False, op=None):
         if other is NaT:
             return
-        if self.freqstr != other.freqstr:
+        if self.freqstr != other.freqstr:  # TODO: dont raise for eq/ne comparison?
             raise raise_on_incompatible(self, other)
 
     # --------------------------------------------------------------------
diff --git a/pandas/core/arrays/timedeltas.py b/pandas/core/arrays/timedeltas.py
index 8c93dca783113..c5a45509857a4 100644
--- a/pandas/core/arrays/timedeltas.py
+++ b/pandas/core/arrays/timedeltas.py
@@ -282,7 +282,7 @@ def _unbox_scalar(self, value):
     def _scalar_from_string(self, value):
         return Timedelta(value)
 
-    def _check_compatible_with(self, other, setitem: bool = False):
+    def _check_compatible_with(self, other, setitem: bool = False, op=None):
         # we don't have anything to validate.
         pass
 
diff --git a/pandas/tests/arithmetic/test_datetime64.py b/pandas/tests/arithmetic/test_datetime64.py
index 56c5647d865d3..57f44f4450ee1 100644
--- a/pandas/tests/arithmetic/test_datetime64.py
+++ b/pandas/tests/arithmetic/test_datetime64.py
@@ -539,25 +539,43 @@ def test_comparison_tzawareness_compat(self, op, box_df_fail):
         dr = tm.box_expected(dr, box)
         dz = tm.box_expected(dz, box)
 
-        msg = "Cannot compare tz-naive and tz-aware"
-        with pytest.raises(TypeError, match=msg):
-            op(dr, dz)
+        if op not in [operator.eq, operator.ne]:
+            msg = "Cannot compare tz-naive and tz-aware"
+            with pytest.raises(TypeError, match=msg):
+                op(dr, dz)
 
-        # FIXME: DataFrame case fails to raise for == and !=, wrong
-        #  message for inequalities
-        with pytest.raises(TypeError, match=msg):
-            op(dr, list(dz))
-        with pytest.raises(TypeError, match=msg):
-            op(dr, np.array(list(dz), dtype=object))
-        with pytest.raises(TypeError, match=msg):
-            op(dz, dr)
+            # FIXME: DataFrame case fails to raise for == and !=, wrong
+            #  message for inequalities
+            with pytest.raises(TypeError, match=msg):
+                op(dr, list(dz))
+            with pytest.raises(TypeError, match=msg):
+                op(dr, np.array(list(dz), dtype=object))
+            with pytest.raises(TypeError, match=msg):
+                op(dz, dr)
 
-        # FIXME: DataFrame case fails to raise for == and !=, wrong
-        #  message for inequalities
-        with pytest.raises(TypeError, match=msg):
-            op(dz, list(dr))
-        with pytest.raises(TypeError, match=msg):
-            op(dz, np.array(list(dr), dtype=object))
+            # FIXME: DataFrame case fails to raise for == and !=, wrong
+            #  message for inequalities
+            with pytest.raises(TypeError, match=msg):
+                op(dz, list(dr))
+            with pytest.raises(TypeError, match=msg):
+                op(dz, np.array(list(dr), dtype=object))
+
+        elif op is operator.eq:
+
+            assert not np.any(op(dr, dz))
+            assert not np.any(op(dr, list(dz)))
+            assert not np.any(op(dr, np.array(list(dz), dtype=object)))
+            assert not np.any(op(dz, dr))
+            assert not np.any(op(dz, list(dr)))
+            assert not np.any(op(dz, np.array(list(dr), dtype=object)))
+
+        else:
+            assert np.all(op(dr, dz))
+            assert np.all(op(dr, list(dz)))
+            assert np.all(op(dr, np.array(list(dz), dtype=object)))
+            assert np.all(op(dz, dr))
+            assert np.all(op(dz, list(dr)))
+            assert np.all(op(dz, np.array(list(dr), dtype=object)))
 
         # The aware==aware and naive==naive comparisons should *not* raise
         assert np.all(dr == dr)
@@ -589,17 +607,27 @@ def test_comparison_tzawareness_compat_scalars(self, op, box_with_array):
         ts_tz = pd.Timestamp("2000-03-14 01:59", tz="Europe/Amsterdam")
 
         assert np.all(dr > ts)
-        msg = "Cannot compare tz-naive and tz-aware"
-        with pytest.raises(TypeError, match=msg):
-            op(dr, ts_tz)
 
-        assert np.all(dz > ts_tz)
-        with pytest.raises(TypeError, match=msg):
-            op(dz, ts)
+        if op not in [operator.eq, operator.ne]:
+            msg = "Cannot compare tz-naive and tz-aware"
+            with pytest.raises(TypeError, match=msg):
+                op(dr, ts_tz)
 
-        # GH#12601: Check comparison against Timestamps and DatetimeIndex
-        with pytest.raises(TypeError, match=msg):
-            op(ts, dz)
+            assert np.all(dz > ts_tz)
+            with pytest.raises(TypeError, match=msg):
+                op(dz, ts)
+
+            # GH#12601: Check comparison against Timestamps and DatetimeIndex
+            with pytest.raises(TypeError, match=msg):
+                op(ts, dz)
+        elif op is operator.eq:
+            assert not np.any(op(dr, ts_tz))
+            assert not np.any(op(dz, ts))
+            assert not np.any(op(ts, dz))
+        else:
+            assert np.all(op(dr, ts_tz))
+            assert np.all(op(dz, ts))
+            assert np.all(op(ts, dz))
 
     @pytest.mark.parametrize(
         "op",
@@ -621,11 +649,19 @@ def test_scalar_comparison_tzawareness(
         dti = pd.date_range("2016-01-01", periods=2, tz=tz)
 
         dtarr = tm.box_expected(dti, box_with_array)
-        msg = "Cannot compare tz-naive and tz-aware"
-        with pytest.raises(TypeError, match=msg):
-            op(dtarr, other)
-        with pytest.raises(TypeError, match=msg):
-            op(other, dtarr)
+        if op not in [operator.eq, operator.ne]:
+            msg = "Cannot compare tz-naive and tz-aware"
+            with pytest.raises(TypeError, match=msg):
+                op(dtarr, other)
+            with pytest.raises(TypeError, match=msg):
+                op(other, dtarr)
+        elif op is operator.eq:
+
+            assert not np.any(op(dtarr, other))
+            assert not np.any(op(other, dtarr))
+        else:
+            assert np.all(op(dtarr, other))
+            assert np.all(op(other, dtarr))
 
     @pytest.mark.parametrize(
         "op",
@@ -725,10 +761,9 @@ def test_dti_cmp_object_dtype(self):
         tm.assert_numpy_array_equal(result, expected)
 
         other = dti.tz_localize(None)
-        msg = "Cannot compare tz-naive and tz-aware"
-        with pytest.raises(TypeError, match=msg):
-            # tzawareness failure
-            dti != other
+        expected = np.ones(dti.shape, dtype=bool)
+        result = dti != other
+        tm.assert_numpy_array_equal(result, expected)
 
         other = np.array(list(dti[:5]) + [Timedelta(days=1)] * 5)
         result = dti == other
diff --git a/pandas/tests/reductions/test_reductions.py b/pandas/tests/reductions/test_reductions.py
index fa62d5d8c4983..8f7224d7ab44e 100644
--- a/pandas/tests/reductions/test_reductions.py
+++ b/pandas/tests/reductions/test_reductions.py
@@ -52,17 +52,15 @@ class TestReductions:
     @pytest.mark.parametrize("obj", objs)
     def test_ops(self, opname, obj):
         result = getattr(obj, opname)()
-        if not isinstance(obj, PeriodIndex):
+        if isinstance(obj, DatetimeIndex):
+            expected = getattr(obj.values, opname)()
+            expected = obj._box_func(expected)
+        elif not isinstance(obj, PeriodIndex):
             expected = getattr(obj.values, opname)()
         else:
             expected = pd.Period(ordinal=getattr(obj.asi8, opname)(), freq=obj.freq)
-        try:
-            assert result == expected
-        except TypeError:
-            # comparing tz-aware series with np.array results in
-            # TypeError
-            expected = expected.astype("M8[ns]").astype("int64")
-            assert result.value == expected
+
+        assert result == expected
 
     @pytest.mark.parametrize("opname", ["max", "min"])
     @pytest.mark.parametrize(
diff --git a/pandas/tests/scalar/timestamp/test_comparisons.py b/pandas/tests/scalar/timestamp/test_comparisons.py
index 27aef8c4a9eb7..bc8bed477fd2f 100644
--- a/pandas/tests/scalar/timestamp/test_comparisons.py
+++ b/pandas/tests/scalar/timestamp/test_comparisons.py
@@ -138,11 +138,9 @@ def test_cant_compare_tz_naive_w_aware(self, utc_fixture):
         a = Timestamp("3/12/2012")
         b = Timestamp("3/12/2012", tz=utc_fixture)
 
+        assert not a == b
+        assert a != b
         msg = "Cannot compare tz-naive and tz-aware timestamps"
-        with pytest.raises(TypeError, match=msg):
-            a == b
-        with pytest.raises(TypeError, match=msg):
-            a != b
         with pytest.raises(TypeError, match=msg):
             a < b
         with pytest.raises(TypeError, match=msg):
@@ -152,10 +150,8 @@ def test_cant_compare_tz_naive_w_aware(self, utc_fixture):
         with pytest.raises(TypeError, match=msg):
             a >= b
 
-        with pytest.raises(TypeError, match=msg):
-            b == a
-        with pytest.raises(TypeError, match=msg):
-            b != a
+        assert not b == a
+        assert b != a
         with pytest.raises(TypeError, match=msg):
             b < a
         with pytest.raises(TypeError, match=msg):