diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index c482eae35b313..8fa2c734092f4 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -101,6 +101,8 @@ class DatetimeLikeArrayMixin(OpsMixin, NDArrayBackedExtensionArray):
         _generate_range
     """
 
+    # _infer_matches -> which infer_dtype strings are close enough to our own
+    _infer_matches: Tuple[str, ...]
     _is_recognized_dtype: Callable[[DtypeObj], bool]
     _recognized_scalars: Tuple[Type, ...]
     _data: np.ndarray
diff --git a/pandas/core/arrays/datetimes.py b/pandas/core/arrays/datetimes.py
index 7c6b38d9114ab..ce70f929cc79d 100644
--- a/pandas/core/arrays/datetimes.py
+++ b/pandas/core/arrays/datetimes.py
@@ -154,6 +154,7 @@ class DatetimeArray(dtl.TimelikeOps, dtl.DatelikeOps):
     _scalar_type = Timestamp
     _recognized_scalars = (datetime, np.datetime64)
     _is_recognized_dtype = is_datetime64_any_dtype
+    _infer_matches = ("datetime", "datetime64", "date")
 
     # define my properties & methods for delegation
     _bool_ops = [
diff --git a/pandas/core/arrays/period.py b/pandas/core/arrays/period.py
index 80882acceb56a..50ed526cf01e9 100644
--- a/pandas/core/arrays/period.py
+++ b/pandas/core/arrays/period.py
@@ -124,6 +124,7 @@ class PeriodArray(PeriodMixin, dtl.DatelikeOps):
     _scalar_type = Period
     _recognized_scalars = (Period,)
     _is_recognized_dtype = is_period_dtype
+    _infer_matches = ("period",)
 
     # Names others delegate to us
     _other_ops: List[str] = []
diff --git a/pandas/core/arrays/timedeltas.py b/pandas/core/arrays/timedeltas.py
index 035e6e84c6ec8..998117cc49d50 100644
--- a/pandas/core/arrays/timedeltas.py
+++ b/pandas/core/arrays/timedeltas.py
@@ -104,6 +104,7 @@ class TimedeltaArray(dtl.TimelikeOps):
     _scalar_type = Timedelta
     _recognized_scalars = (timedelta, np.timedelta64, Tick)
     _is_recognized_dtype = is_timedelta64_dtype
+    _infer_matches = ("timedelta", "timedelta64")
 
     __array_priority__ = 1000
     # define my properties & methods for delegation
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index d0f818410f96a..57f6a8ea0cca5 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -157,16 +157,8 @@ def equals(self, other: object) -> bool:
         elif other.dtype.kind in ["f", "i", "u", "c"]:
             return False
         elif not isinstance(other, type(self)):
-            inferrable = [
-                "timedelta",
-                "timedelta64",
-                "datetime",
-                "datetime64",
-                "date",
-                "period",
-            ]
-
             should_try = False
+            inferrable = self._data._infer_matches
             if other.dtype == object:
                 should_try = other.inferred_type in inferrable
             elif is_categorical_dtype(other.dtype):
@@ -648,6 +640,9 @@ def _has_complex_internals(self) -> bool:
         # used to avoid libreduction code paths, which raise or require conversion
         return False
 
+    def is_type_compatible(self, kind: str) -> bool:
+        return kind in self._data._infer_matches
+
     # --------------------------------------------------------------------
     # Set Operation Methods
 
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 1dd3eb1017eca..b39a36d95d27b 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -814,9 +814,6 @@ def slice_indexer(self, start=None, end=None, step=None, kind=None):
 
     # --------------------------------------------------------------------
 
-    def is_type_compatible(self, kind: str) -> bool:
-        return kind == self.inferred_type or kind == "datetime"
-
     @property
     def inferred_type(self) -> str:
         # b/c datetime is represented as microseconds since the epoch, make
diff --git a/pandas/core/indexes/timedeltas.py b/pandas/core/indexes/timedeltas.py
index 27e090f450cd8..f44a1701bfa9b 100644
--- a/pandas/core/indexes/timedeltas.py
+++ b/pandas/core/indexes/timedeltas.py
@@ -229,9 +229,6 @@ def _maybe_cast_slice_bound(self, label, side: str, kind):
 
     # -------------------------------------------------------------------
 
-    def is_type_compatible(self, kind: str) -> bool:
-        return kind == self.inferred_type or kind == "timedelta"
-
     @property
     def inferred_type(self) -> str:
         return "timedelta64"