diff --git a/pandas/core/indexes/category.py b/pandas/core/indexes/category.py
index 06df8f85cded7..4b79ac9faf769 100644
--- a/pandas/core/indexes/category.py
+++ b/pandas/core/indexes/category.py
@@ -224,7 +224,6 @@ def _simple_new(cls, values: Categorical, name: Label = None):
         result._cache = {}
 
         result._reset_identity()
-        result._no_setting_name = False
         return result
 
     # --------------------------------------------------------------------
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 40a6086f69f85..88383cb7bbb6a 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -2,7 +2,7 @@
 Base and utility classes for tseries type pandas objects.
 """
 from datetime import datetime
-from typing import TYPE_CHECKING, Any, List, Optional, Tuple, TypeVar, Union, cast
+from typing import TYPE_CHECKING, Any, List, Optional, Tuple, Type, TypeVar, Union, cast
 
 import numpy as np
 
@@ -88,6 +88,7 @@ class DatetimeIndexOpsMixin(NDArrayBackedExtensionIndex):
 
     _can_hold_strings = False
     _data: Union[DatetimeArray, TimedeltaArray, PeriodArray]
+    _data_cls: Union[Type[DatetimeArray], Type[TimedeltaArray], Type[PeriodArray]]
     freq: Optional[BaseOffset]
     freqstr: Optional[str]
     _resolution_obj: Resolution
@@ -100,6 +101,25 @@ class DatetimeIndexOpsMixin(NDArrayBackedExtensionIndex):
     )
     _hasnans = hasnans  # for index / array -agnostic code
 
+    @classmethod
+    def _simple_new(
+        cls,
+        values: Union[DatetimeArray, TimedeltaArray, PeriodArray],
+        name: Label = None,
+    ):
+        assert isinstance(values, cls._data_cls), type(values)
+
+        result = object.__new__(cls)
+        result._data = values
+        result._name = name
+        result._cache = {}
+
+        # For groupby perf. See note in indexes/base about _index_data
+        result._index_data = values._data
+
+        result._reset_identity()
+        return result
+
     @property
     def _is_all_dates(self) -> bool:
         return True
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index a1a1a83555f95..73f977c3dadf9 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -14,7 +14,7 @@
     to_offset,
 )
 from pandas._libs.tslibs.offsets import prefix_mapping
-from pandas._typing import DtypeObj, Label
+from pandas._typing import DtypeObj
 from pandas.errors import InvalidIndexError
 from pandas.util._decorators import cache_readonly, doc
 
@@ -220,6 +220,7 @@ class DatetimeIndex(DatetimeTimedeltaMixin):
 
     _typ = "datetimeindex"
 
+    _data_cls = DatetimeArray
     _engine_type = libindex.DatetimeEngine
     _supports_partial_string_indexing = True
 
@@ -319,20 +320,6 @@ def __new__(
         subarr = cls._simple_new(dtarr, name=name)
         return subarr
 
-    @classmethod
-    def _simple_new(cls, values: DatetimeArray, name: Label = None):
-        assert isinstance(values, DatetimeArray), type(values)
-
-        result = object.__new__(cls)
-        result._data = values
-        result.name = name
-        result._cache = {}
-        result._no_setting_name = False
-        # For groupby perf. See note in indexes/base about _index_data
-        result._index_data = values._data
-        result._reset_identity()
-        return result
-
     # --------------------------------------------------------------------
 
     @cache_readonly
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index 2aec86c9cdfae..bfbb4a00638fc 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -239,7 +239,6 @@ def _simple_new(cls, array: IntervalArray, name: Label = None):
         result._data = array
         result.name = name
         result._cache = {}
-        result._no_setting_name = False
         result._reset_identity()
         return result
 
diff --git a/pandas/core/indexes/period.py b/pandas/core/indexes/period.py
index 44c20ad0de848..c0a3b95499b3d 100644
--- a/pandas/core/indexes/period.py
+++ b/pandas/core/indexes/period.py
@@ -146,6 +146,7 @@ class PeriodIndex(DatetimeIndexOpsMixin, Int64Index):
     _data: PeriodArray
     freq: BaseOffset
 
+    _data_cls = PeriodArray
     _engine_type = libindex.PeriodEngine
     _supports_partial_string_indexing = True
 
@@ -244,28 +245,6 @@ def __new__(
 
         return cls._simple_new(data, name=name)
 
-    @classmethod
-    def _simple_new(cls, values: PeriodArray, name: Label = None):
-        """
-        Create a new PeriodIndex.
-
-        Parameters
-        ----------
-        values : PeriodArray
-            Values that can be converted to a PeriodArray without inference
-            or coercion.
-        """
-        assert isinstance(values, PeriodArray), type(values)
-
-        result = object.__new__(cls)
-        result._data = values
-        # For groupby perf. See note in indexes/base about _index_data
-        result._index_data = values._data
-        result.name = name
-        result._cache = {}
-        result._reset_identity()
-        return result
-
     # ------------------------------------------------------------------------
     # Data
 
diff --git a/pandas/core/indexes/timedeltas.py b/pandas/core/indexes/timedeltas.py
index cf5fa4bbb3d75..b6c626197eaf9 100644
--- a/pandas/core/indexes/timedeltas.py
+++ b/pandas/core/indexes/timedeltas.py
@@ -2,7 +2,7 @@
 
 from pandas._libs import index as libindex, lib
 from pandas._libs.tslibs import Timedelta, to_offset
-from pandas._typing import DtypeObj, Label
+from pandas._typing import DtypeObj
 from pandas.errors import InvalidIndexError
 from pandas.util._decorators import doc
 
@@ -103,6 +103,7 @@ class TimedeltaIndex(DatetimeTimedeltaMixin):
 
     _typ = "timedeltaindex"
 
+    _data_cls = TimedeltaArray
     _engine_type = libindex.TimedeltaEngine
 
     _comparables = ["name", "freq"]
@@ -156,20 +157,6 @@ def __new__(
         )
         return cls._simple_new(tdarr, name=name)
 
-    @classmethod
-    def _simple_new(cls, values: TimedeltaArray, name: Label = None):
-        assert isinstance(values, TimedeltaArray)
-
-        result = object.__new__(cls)
-        result._data = values
-        result._name = name
-        result._cache = {}
-        # For groupby perf. See note in indexes/base about _index_data
-        result._index_data = values._data
-
-        result._reset_identity()
-        return result
-
     # -------------------------------------------------------------------
     # Rendering Methods