diff --git a/pandas/core/arrays/timedeltas.py b/pandas/core/arrays/timedeltas.py
index f85e3e716bbf9..0853664c766bb 100644
--- a/pandas/core/arrays/timedeltas.py
+++ b/pandas/core/arrays/timedeltas.py
@@ -23,6 +23,7 @@
 from pandas.core.dtypes.common import (
     DT64NS_DTYPE,
     TD64NS_DTYPE,
+    is_categorical_dtype,
     is_dtype_equal,
     is_float_dtype,
     is_integer_dtype,
@@ -940,6 +941,9 @@ def sequence_to_td64ns(data, copy=False, unit=None, errors="raise"):
         data = data._data
     elif isinstance(data, IntegerArray):
         data = data.to_numpy("int64", na_value=tslibs.iNaT)
+    elif is_categorical_dtype(data.dtype):
+        data = data.categories.take(data.codes, fill_value=NaT)._values
+        copy = False
 
     # Convert whatever we have into timedelta64[ns] dtype
     if is_object_dtype(data.dtype) or is_string_dtype(data.dtype):
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 5821ff0aca3c2..5baa103a25d51 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, tzinfo
-from typing import Any, List, Optional, TypeVar, Union, cast
+from typing import TYPE_CHECKING, Any, List, Optional, TypeVar, Union, cast
 
 import numpy as np
 
@@ -16,6 +16,7 @@
 from pandas.core.dtypes.common import (
     ensure_int64,
     is_bool_dtype,
+    is_categorical_dtype,
     is_dtype_equal,
     is_integer,
     is_list_like,
@@ -41,6 +42,9 @@
 from pandas.core.ops import get_op_result_name
 from pandas.core.tools.timedeltas import to_timedelta
 
+if TYPE_CHECKING:
+    from pandas import CategoricalIndex
+
 _index_doc_kwargs = dict(ibase._index_doc_kwargs)
 
 _T = TypeVar("_T", bound="DatetimeIndexOpsMixin")
@@ -137,14 +141,31 @@ def equals(self, other: object) -> bool:
         elif other.dtype.kind in ["f", "i", "u", "c"]:
             return False
         elif not isinstance(other, type(self)):
-            try:
-                other = type(self)(other)
-            except (ValueError, TypeError, OverflowError):
-                # e.g.
-                #  ValueError -> cannot parse str entry, or OutOfBoundsDatetime
-                #  TypeError  -> trying to convert IntervalIndex to DatetimeIndex
-                #  OverflowError -> Index([very_large_timedeltas])
-                return False
+            inferrable = [
+                "timedelta",
+                "timedelta64",
+                "datetime",
+                "datetime64",
+                "date",
+                "period",
+            ]
+
+            should_try = False
+            if other.dtype == object:
+                should_try = other.inferred_type in inferrable
+            elif is_categorical_dtype(other.dtype):
+                other = cast("CategoricalIndex", other)
+                should_try = other.categories.inferred_type in inferrable
+
+            if should_try:
+                try:
+                    other = type(self)(other)
+                except (ValueError, TypeError, OverflowError):
+                    # e.g.
+                    #  ValueError -> cannot parse str entry, or OutOfBoundsDatetime
+                    #  TypeError  -> trying to convert IntervalIndex to DatetimeIndex
+                    #  OverflowError -> Index([very_large_timedeltas])
+                    return False
 
         if not is_dtype_equal(self.dtype, other.dtype):
             # have different timezone
diff --git a/pandas/tests/indexes/datetimelike.py b/pandas/tests/indexes/datetimelike.py
index df857cce05bbb..be8ca61f1a730 100644
--- a/pandas/tests/indexes/datetimelike.py
+++ b/pandas/tests/indexes/datetimelike.py
@@ -116,6 +116,20 @@ def test_not_equals_numeric(self):
         assert not index.equals(pd.Index(index.asi8.astype("u8")))
         assert not index.equals(pd.Index(index.asi8).astype("f8"))
 
+    def test_equals(self):
+        index = self.create_index()
+
+        assert index.equals(index.astype(object))
+        assert index.equals(pd.CategoricalIndex(index))
+        assert index.equals(pd.CategoricalIndex(index.astype(object)))
+
+    def test_not_equals_strings(self):
+        index = self.create_index()
+
+        other = pd.Index([str(x) for x in index], dtype=object)
+        assert not index.equals(other)
+        assert not index.equals(pd.CategoricalIndex(other))
+
     def test_where_cast_str(self):
         index = self.create_index()
 
diff --git a/pandas/tests/indexes/timedeltas/test_constructors.py b/pandas/tests/indexes/timedeltas/test_constructors.py
index 41e4e220c999c..09344bb5054f6 100644
--- a/pandas/tests/indexes/timedeltas/test_constructors.py
+++ b/pandas/tests/indexes/timedeltas/test_constructors.py
@@ -238,3 +238,15 @@ def test_explicit_none_freq(self):
 
         result = TimedeltaIndex(tdi._data, freq=None)
         assert result.freq is None
+
+    def test_from_categorical(self):
+        tdi = timedelta_range(1, periods=5)
+
+        cat = pd.Categorical(tdi)
+
+        result = TimedeltaIndex(cat)
+        tm.assert_index_equal(result, tdi)
+
+        ci = pd.CategoricalIndex(tdi)
+        result = TimedeltaIndex(ci)
+        tm.assert_index_equal(result, tdi)