diff --git a/pandas/_libs/tslibs/offsets.pyi b/pandas/_libs/tslibs/offsets.pyi
index c3d550c7a5ba9..34c5c661f6ec9 100644
--- a/pandas/_libs/tslibs/offsets.pyi
+++ b/pandas/_libs/tslibs/offsets.pyi
@@ -13,6 +13,7 @@ from typing import (
 
 import numpy as np
 
+from pandas._libs.tslibs.nattype import NaTType
 from pandas._typing import npt
 
 from .timedeltas import Timedelta
@@ -49,6 +50,8 @@ class BaseOffset:
     @overload
     def __radd__(self, other: npt.NDArray[np.object_]) -> npt.NDArray[np.object_]: ...
     @overload
+    def __radd__(self, other: NaTType) -> NaTType: ...
+    @overload
     def __radd__(self: _BaseOffsetT, other: BaseOffset) -> _BaseOffsetT: ...
     @overload
     def __radd__(self, other: _DatetimeT) -> _DatetimeT: ...
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index 9d32bc008eb25..c946fc2ac6e13 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -925,7 +925,7 @@ def _maybe_mask_results(
     # Frequency Properties/Methods
 
     @property
-    def freq(self):
+    def freq(self) -> BaseOffset | None:
         """
         Return the frequency object if it is set, otherwise None.
         """
@@ -1220,7 +1220,9 @@ def _sub_period(self, other: Period) -> npt.NDArray[np.object_]:
         new_i8_data = checked_add_with_arr(
             self.asi8, -other.ordinal, arr_mask=self._isnan
         )
-        new_data = np.array([self.freq.base * x for x in new_i8_data])
+        new_data = np.array(
+            [cast("PeriodArray", self).freq.base * x for x in new_i8_data]
+        )
 
         if self._hasna:
             new_data[self._isnan] = NaT
@@ -1456,8 +1458,9 @@ def __add__(self, other):
             # as is_integer returns True for these
             if not is_period_dtype(self.dtype):
                 raise integer_op_not_supported(self)
-            result = cast("PeriodArray", self)._addsub_int_array_or_scalar(
-                other * self.freq.n, operator.add
+            self_periodarray = cast("PeriodArray", self)
+            result = self_periodarray._addsub_int_array_or_scalar(
+                other * self_periodarray.freq.n, operator.add
             )
 
         # array-like others
@@ -1473,8 +1476,9 @@ def __add__(self, other):
         elif is_integer_dtype(other_dtype):
             if not is_period_dtype(self.dtype):
                 raise integer_op_not_supported(self)
+            # error: Item "None" of "Optional[BaseOffset]" has no attribute "n"
             result = cast("PeriodArray", self)._addsub_int_array_or_scalar(
-                other * self.freq.n, operator.add
+                other * self.freq.n, operator.add  # type: ignore[union-attr]
             )
         else:
             # Includes Categorical, other ExtensionArrays
@@ -1514,8 +1518,9 @@ def __sub__(self, other):
             # as is_integer returns True for these
             if not is_period_dtype(self.dtype):
                 raise integer_op_not_supported(self)
+            # error: Item "None" of "Optional[BaseOffset]" has no attribute "n"
             result = cast("PeriodArray", self)._addsub_int_array_or_scalar(
-                other * self.freq.n, operator.sub
+                other * self.freq.n, operator.sub  # type: ignore[union-attr]
             )
 
         elif isinstance(other, Period):
@@ -1537,8 +1542,9 @@ def __sub__(self, other):
         elif is_integer_dtype(other_dtype):
             if not is_period_dtype(self.dtype):
                 raise integer_op_not_supported(self)
+            # error: Item "None" of "Optional[BaseOffset]" has no attribute "n"
             result = cast("PeriodArray", self)._addsub_int_array_or_scalar(
-                other * self.freq.n, operator.sub
+                other * self.freq.n, operator.sub  # type: ignore[union-attr]
             )
         else:
             # Includes ExtensionArrays, float_dtype
diff --git a/pandas/core/arrays/timedeltas.py b/pandas/core/arrays/timedeltas.py
index 193ab6dc99350..a76491e04b93c 100644
--- a/pandas/core/arrays/timedeltas.py
+++ b/pandas/core/arrays/timedeltas.py
@@ -36,6 +36,7 @@
     npt,
 )
 from pandas.compat.numpy import function as nv
+from pandas.util._decorators import doc
 from pandas.util._validators import validate_endpoints
 
 from pandas.core.dtypes.astype import astype_td64_unit_conversion
@@ -143,6 +144,23 @@ def _box_func(self, x: np.timedelta64) -> Timedelta | NaTType:
             return NaT
         return Timedelta._from_value_and_reso(y, reso=self._reso)
 
+    # error: Decorated property not supported
+    @property  # type: ignore[misc]
+    @doc(dtl.DatetimeLikeArrayMixin.freq)
+    def freq(self) -> Tick | None:
+        # error: Incompatible return value type (got "Optional[BaseOffset]", expected
+        # "Optional[Tick]")
+        return self._freq  # type: ignore[return-value]
+
+    @freq.setter
+    def freq(self, value) -> None:
+        # python doesn't support super().freq = value (any mypy has some
+        # issue with the workaround)
+        # error: overloaded function has no attribute "fset"
+        super(TimedeltaArray, TimedeltaArray).freq.fset(  # type: ignore[attr-defined]
+            self, value
+        )
+
     @property
     # error: Return type "dtype" of "dtype" incompatible with return type
     # "ExtensionDtype" in supertype "ExtensionArray"
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 6867ef936d45e..ab5fc8f06f5ab 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -3,6 +3,10 @@
 """
 from __future__ import annotations
 
+from abc import (
+    ABC,
+    abstractmethod,
+)
 from datetime import datetime
 import inspect
 from typing import (
@@ -30,6 +34,7 @@
     parsing,
     to_offset,
 )
+from pandas._typing import npt
 from pandas.compat.numpy import function as nv
 from pandas.util._decorators import (
     Appender,
@@ -59,10 +64,7 @@
     Index,
     _index_shared_docs,
 )
-from pandas.core.indexes.extension import (
-    NDArrayBackedExtensionIndex,
-    inherit_names,
-)
+from pandas.core.indexes.extension import NDArrayBackedExtensionIndex
 from pandas.core.indexes.range import RangeIndex
 from pandas.core.tools.timedeltas import to_timedelta
 
@@ -75,13 +77,7 @@
 _TDT = TypeVar("_TDT", bound="DatetimeTimedeltaMixin")
 
 
-@inherit_names(
-    ["inferred_freq", "_resolution_obj", "resolution"],
-    DatetimeLikeArrayMixin,
-    cache=True,
-)
-@inherit_names(["mean", "asi8", "freq", "freqstr"], DatetimeLikeArrayMixin)
-class DatetimeIndexOpsMixin(NDArrayBackedExtensionIndex):
+class DatetimeIndexOpsMixin(NDArrayBackedExtensionIndex, ABC):
     """
     Common ops mixin to support a unified interface datetimelike Index.
     """
@@ -89,9 +85,41 @@ class DatetimeIndexOpsMixin(NDArrayBackedExtensionIndex):
     _is_numeric_dtype = False
     _can_hold_strings = False
     _data: DatetimeArray | TimedeltaArray | PeriodArray
-    freq: BaseOffset | None
-    freqstr: str | None
-    _resolution_obj: Resolution
+
+    # ------------------------------------------------------------------------
+
+    @doc(DatetimeLikeArrayMixin.mean)
+    def mean(self, *, skipna: bool = True, axis: int | None = 0):
+        return self._data.mean(skipna=skipna, axis=axis)
+
+    # error: Decorated property not supported
+    @property  # type: ignore[misc]
+    @doc(DatetimeLikeArrayMixin.asi8)
+    def asi8(self) -> npt.NDArray[np.int64]:
+        return self._data.asi8
+
+    # error: Decorated property not supported
+    @property  # type: ignore[misc]
+    @doc(DatetimeLikeArrayMixin.freq)
+    def freq(self) -> BaseOffset | None:
+        return self._data.freq
+
+    # error: Decorated property not supported
+    @property  # type: ignore[misc]
+    @doc(DatetimeLikeArrayMixin.freqstr)
+    def freqstr(self) -> str | None:
+        return self._data.freqstr
+
+    @cache_readonly
+    @abstractmethod
+    def _resolution_obj(self) -> Resolution:
+        ...
+
+    # error: Decorated property not supported
+    @cache_readonly  # type: ignore[misc]
+    @doc(DatetimeLikeArrayMixin.resolution)
+    def resolution(self) -> str:
+        return self._data.resolution
 
     # ------------------------------------------------------------------------
 
@@ -373,7 +401,7 @@ def _maybe_cast_listlike_indexer(self, keyarr):
         return Index(res, dtype=res.dtype)
 
 
-class DatetimeTimedeltaMixin(DatetimeIndexOpsMixin):
+class DatetimeTimedeltaMixin(DatetimeIndexOpsMixin, ABC):
     """
     Mixin class for methods shared by DatetimeIndex and TimedeltaIndex,
     but not PeriodIndex
@@ -408,6 +436,23 @@ def values(self) -> np.ndarray:
         # NB: For Datetime64TZ this is lossy
         return self._data._ndarray
 
+    # error: Decorated property not supported
+    @property  # type: ignore[misc]
+    @doc(DatetimeLikeArrayMixin.freq)
+    def freq(self) -> BaseOffset | None:
+        # needed to define the setter (same as in DatetimeIndexOpsMixin)
+        return self._data.freq
+
+    @freq.setter
+    def freq(self, value) -> None:
+        self._data.freq = value
+
+    # error: Decorated property not supported
+    @cache_readonly  # type: ignore[misc]
+    @doc(DatetimeLikeArrayMixin.inferred_freq)
+    def inferred_freq(self) -> str | None:
+        return self._data.inferred_freq
+
     # --------------------------------------------------------------------
     # Set Operation Methods
 
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 80138c25b0c27..d6c5d81a8fae1 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -121,7 +121,7 @@ def _new_DatetimeIndex(cls, d):
     DatetimeArray,
     wrap=True,
 )
-@inherit_names(["is_normalized", "_resolution_obj"], DatetimeArray, cache=True)
+@inherit_names(["is_normalized"], DatetimeArray, cache=True)
 @inherit_names(
     [
         "tz",
@@ -261,7 +261,6 @@ def _engine_type(self) -> type[libindex.DatetimeEngine]:
         return libindex.DatetimeEngine
 
     _data: DatetimeArray
-    inferred_freq: str | None
     tz: tzinfo | None
 
     # --------------------------------------------------------------------
@@ -308,6 +307,10 @@ def isocalendar(self) -> DataFrame:
         df = self._data.isocalendar()
         return df.set_index(self)
 
+    @cache_readonly
+    def _resolution_obj(self) -> Resolution:
+        return self._data._resolution_obj
+
     # --------------------------------------------------------------------
     # Constructors
 
diff --git a/pandas/core/indexes/period.py b/pandas/core/indexes/period.py
index fedcba7aa9644..e2741aa185051 100644
--- a/pandas/core/indexes/period.py
+++ b/pandas/core/indexes/period.py
@@ -167,8 +167,7 @@ def _engine_type(self) -> type[libindex.PeriodEngine]:
         return libindex.PeriodEngine
 
     @cache_readonly
-    # Signature of "_resolution_obj" incompatible with supertype "DatetimeIndexOpsMixin"
-    def _resolution_obj(self) -> Resolution:  # type: ignore[override]
+    def _resolution_obj(self) -> Resolution:
         # for compat with DatetimeIndex
         return self.dtype._resolution_obj
 
@@ -393,7 +392,7 @@ def is_full(self) -> bool:
         if not self.is_monotonic_increasing:
             raise ValueError("Index is not monotonic")
         values = self.asi8
-        return ((values[1:] - values[:-1]) < 2).all()
+        return ((values[1:] - values[:-1]) < 2).all().item()
 
     @property
     def inferred_type(self) -> str:
diff --git a/pandas/core/indexes/timedeltas.py b/pandas/core/indexes/timedeltas.py
index 12a8f2c0d5a9d..c7bcdac9c6739 100644
--- a/pandas/core/indexes/timedeltas.py
+++ b/pandas/core/indexes/timedeltas.py
@@ -112,6 +112,13 @@ def _engine_type(self) -> type[libindex.TimedeltaEngine]:
     # Use base class method instead of DatetimeTimedeltaMixin._get_string_slice
     _get_string_slice = Index._get_string_slice
 
+    # error: Return type "None" of "_resolution_obj" incompatible with return type
+    # "Resolution" in supertype "DatetimeIndexOpsMixin"
+    @property
+    def _resolution_obj(self) -> None:  # type: ignore[override]
+        # not used but need to implement it because it is an abstract method
+        return None
+
     # -------------------------------------------------------------------
     # Constructors
 
diff --git a/pandas/tests/indexes/period/test_freq_attr.py b/pandas/tests/indexes/period/test_freq_attr.py
index e1ecffa4982bd..71115929121bc 100644
--- a/pandas/tests/indexes/period/test_freq_attr.py
+++ b/pandas/tests/indexes/period/test_freq_attr.py
@@ -20,7 +20,7 @@ def test_freq_setter_deprecated(self):
 
         # warning for setter
         msg = (
-            "property 'freq' of 'PeriodArray' object has no setter"
+            "property 'freq' of 'PeriodIndex' object has no setter"
             if PY311
             else "can't set attribute"
         )