diff --git a/pandas/core/base.py b/pandas/core/base.py
index 048831a74d1a5..b0993bbc619dc 100644
--- a/pandas/core/base.py
+++ b/pandas/core/base.py
@@ -52,7 +52,10 @@
     remove_na_arraylike,
 )
 
-from pandas.core import algorithms
+from pandas.core import (
+    algorithms,
+    ops,
+)
 from pandas.core.accessor import DirNamesMixin
 from pandas.core.algorithms import (
     duplicated,
@@ -61,7 +64,11 @@
 )
 from pandas.core.arraylike import OpsMixin
 from pandas.core.arrays import ExtensionArray
-from pandas.core.construction import create_series_with_explicit_dtype
+from pandas.core.construction import (
+    create_series_with_explicit_dtype,
+    ensure_wrapped_if_datetimelike,
+    extract_array,
+)
 import pandas.core.nanops as nanops
 
 if TYPE_CHECKING:
@@ -1238,3 +1245,23 @@ def _duplicated(
         self, keep: Literal["first", "last", False] = "first"
     ) -> npt.NDArray[np.bool_]:
         return duplicated(self._values, keep=keep)
+
+    def _arith_method(self, other, op):
+        res_name = ops.get_op_result_name(self, other)
+
+        lvalues = self._values
+        rvalues = extract_array(other, extract_numpy=True, extract_range=True)
+        rvalues = ops.maybe_prepare_scalar_for_op(rvalues, lvalues.shape)
+        rvalues = ensure_wrapped_if_datetimelike(rvalues)
+
+        with np.errstate(all="ignore"):
+            result = ops.arithmetic_op(lvalues, rvalues, op)
+
+        return self._construct_result(result, name=res_name)
+
+    def _construct_result(self, result, name):
+        """
+        Construct an appropriately-wrapped result from the ArrayLike result
+        of an arithmetic-like operation.
+        """
+        raise AbstractMethodError(self)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 6c33510ddf1bc..a1c411031a465 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -6364,7 +6364,10 @@ def _cmp_method(self, other, op):
                     arr[self.isna()] = False
                 return arr
             elif op in {operator.ne, operator.lt, operator.gt}:
-                return np.zeros(len(self), dtype=bool)
+                arr = np.zeros(len(self), dtype=bool)
+                if self._can_hold_na and not isinstance(self, ABCMultiIndex):
+                    arr[self.isna()] = True
+                return arr
 
         if isinstance(other, (np.ndarray, Index, ABCSeries, ExtensionArray)) and len(
             self
@@ -6381,6 +6384,9 @@ def _cmp_method(self, other, op):
             with np.errstate(all="ignore"):
                 result = op(self._values, other)
 
+        elif isinstance(self._values, ExtensionArray):
+            result = op(self._values, other)
+
         elif is_object_dtype(self.dtype) and not isinstance(self, ABCMultiIndex):
             # don't pass MultiIndex
             with np.errstate(all="ignore"):
@@ -6392,17 +6398,26 @@ def _cmp_method(self, other, op):
 
         return result
 
-    def _arith_method(self, other, op):
-        """
-        Wrapper used to dispatch arithmetic operations.
-        """
+    def _construct_result(self, result, name):
+        if isinstance(result, tuple):
+            return (
+                Index._with_infer(result[0], name=name),
+                Index._with_infer(result[1], name=name),
+            )
+        return Index._with_infer(result, name=name)
 
-        from pandas import Series
+    def _arith_method(self, other, op):
+        if (
+            isinstance(other, Index)
+            and is_object_dtype(other.dtype)
+            and type(other) is not Index
+        ):
+            # We return NotImplemented for object-dtype index *subclasses* so they have
+            # a chance to implement ops before we unwrap them.
+            # See https://github.com/pandas-dev/pandas/issues/31109
+            return NotImplemented
 
-        result = op(Series(self), other)
-        if isinstance(result, tuple):
-            return (Index._with_infer(result[0]), Index(result[1]))
-        return Index._with_infer(result)
+        return super()._arith_method(other, op)
 
     @final
     def _unary_method(self, op):
diff --git a/pandas/core/indexes/extension.py b/pandas/core/indexes/extension.py
index cea67aa13edd1..44267895a989e 100644
--- a/pandas/core/indexes/extension.py
+++ b/pandas/core/indexes/extension.py
@@ -26,13 +26,9 @@
 
 from pandas.core.dtypes.common import (
     is_dtype_equal,
-    is_object_dtype,
     pandas_dtype,
 )
-from pandas.core.dtypes.generic import (
-    ABCDataFrame,
-    ABCSeries,
-)
+from pandas.core.dtypes.generic import ABCDataFrame
 
 from pandas.core.arrays import (
     Categorical,
@@ -45,7 +41,6 @@
 from pandas.core.arrays.base import ExtensionArray
 from pandas.core.indexers import deprecate_ndim_indexing
 from pandas.core.indexes.base import Index
-from pandas.core.ops import get_op_result_name
 
 if TYPE_CHECKING:
 
@@ -154,94 +149,6 @@ def wrapper(cls):
     return wrapper
 
 
-def _make_wrapped_comparison_op(opname: str):
-    """
-    Create a comparison method that dispatches to ``._data``.
-    """
-
-    def wrapper(self, other):
-        if isinstance(other, ABCSeries):
-            # the arrays defer to Series for comparison ops but the indexes
-            #  don't, so we have to unwrap here.
-            other = other._values
-
-        other = _maybe_unwrap_index(other)
-
-        op = getattr(self._data, opname)
-        return op(other)
-
-    wrapper.__name__ = opname
-    return wrapper
-
-
-def _make_wrapped_arith_op(opname: str):
-    def method(self, other):
-        if (
-            isinstance(other, Index)
-            and is_object_dtype(other.dtype)
-            and type(other) is not Index
-        ):
-            # We return NotImplemented for object-dtype index *subclasses* so they have
-            # a chance to implement ops before we unwrap them.
-            # See https://github.com/pandas-dev/pandas/issues/31109
-            return NotImplemented
-
-        try:
-            meth = getattr(self._data, opname)
-        except AttributeError as err:
-            # e.g. Categorical, IntervalArray
-            cls = type(self).__name__
-            raise TypeError(
-                f"cannot perform {opname} with this index type: {cls}"
-            ) from err
-
-        result = meth(_maybe_unwrap_index(other))
-        return _wrap_arithmetic_op(self, other, result)
-
-    method.__name__ = opname
-    return method
-
-
-def _wrap_arithmetic_op(self, other, result):
-    if result is NotImplemented:
-        return NotImplemented
-
-    if isinstance(result, tuple):
-        # divmod, rdivmod
-        assert len(result) == 2
-        return (
-            _wrap_arithmetic_op(self, other, result[0]),
-            _wrap_arithmetic_op(self, other, result[1]),
-        )
-
-    if not isinstance(result, Index):
-        # Index.__new__ will choose appropriate subclass for dtype
-        result = Index(result)
-
-    res_name = get_op_result_name(self, other)
-    result.name = res_name
-    return result
-
-
-def _maybe_unwrap_index(obj):
-    """
-    If operating against another Index object, we need to unwrap the underlying
-    data before deferring to the DatetimeArray/TimedeltaArray/PeriodArray
-    implementation, otherwise we will incorrectly return NotImplemented.
-
-    Parameters
-    ----------
-    obj : object
-
-    Returns
-    -------
-    unwrapped object
-    """
-    if isinstance(obj, Index):
-        return obj._data
-    return obj
-
-
 class ExtensionIndex(Index):
     """
     Index subclass for indexes backed by ExtensionArray.
@@ -284,30 +191,6 @@ def _simple_new(
         result._reset_identity()
         return result
 
-    __eq__ = _make_wrapped_comparison_op("__eq__")
-    __ne__ = _make_wrapped_comparison_op("__ne__")
-    __lt__ = _make_wrapped_comparison_op("__lt__")
-    __gt__ = _make_wrapped_comparison_op("__gt__")
-    __le__ = _make_wrapped_comparison_op("__le__")
-    __ge__ = _make_wrapped_comparison_op("__ge__")
-
-    __add__ = _make_wrapped_arith_op("__add__")
-    __sub__ = _make_wrapped_arith_op("__sub__")
-    __radd__ = _make_wrapped_arith_op("__radd__")
-    __rsub__ = _make_wrapped_arith_op("__rsub__")
-    __pow__ = _make_wrapped_arith_op("__pow__")
-    __rpow__ = _make_wrapped_arith_op("__rpow__")
-    __mul__ = _make_wrapped_arith_op("__mul__")
-    __rmul__ = _make_wrapped_arith_op("__rmul__")
-    __floordiv__ = _make_wrapped_arith_op("__floordiv__")
-    __rfloordiv__ = _make_wrapped_arith_op("__rfloordiv__")
-    __mod__ = _make_wrapped_arith_op("__mod__")
-    __rmod__ = _make_wrapped_arith_op("__rmod__")
-    __divmod__ = _make_wrapped_arith_op("__divmod__")
-    __rdivmod__ = _make_wrapped_arith_op("__rdivmod__")
-    __truediv__ = _make_wrapped_arith_op("__truediv__")
-    __rtruediv__ = _make_wrapped_arith_op("__rtruediv__")
-
     # ---------------------------------------------------------------------
     # NDarray-Like Methods
 
diff --git a/pandas/core/series.py b/pandas/core/series.py
index 6f964ab09e978..c7756e7f42337 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -104,7 +104,6 @@
 import pandas.core.common as com
 from pandas.core.construction import (
     create_series_with_explicit_dtype,
-    ensure_wrapped_if_datetimelike,
     extract_array,
     is_empty_data,
     sanitize_array,
@@ -5515,18 +5514,8 @@ def _logical_method(self, other, op):
         return self._construct_result(res_values, name=res_name)
 
     def _arith_method(self, other, op):
-        res_name = ops.get_op_result_name(self, other)
         self, other = ops.align_method_SERIES(self, other)
-
-        lvalues = self._values
-        rvalues = extract_array(other, extract_numpy=True, extract_range=True)
-        rvalues = ops.maybe_prepare_scalar_for_op(rvalues, lvalues.shape)
-        rvalues = ensure_wrapped_if_datetimelike(rvalues)
-
-        with np.errstate(all="ignore"):
-            result = ops.arithmetic_op(lvalues, rvalues, op)
-
-        return self._construct_result(result, name=res_name)
+        return base.IndexOpsMixin._arith_method(self, other, op)
 
 
 Series._add_numeric_operations()
diff --git a/pandas/tests/arithmetic/test_datetime64.py b/pandas/tests/arithmetic/test_datetime64.py
index 6b3309ba8ea1b..c0f38a1181026 100644
--- a/pandas/tests/arithmetic/test_datetime64.py
+++ b/pandas/tests/arithmetic/test_datetime64.py
@@ -2146,7 +2146,7 @@ def test_dti_sub_tdi(self, tz_naive_fixture):
         result = dti - tdi.values
         tm.assert_index_equal(result, expected)
 
-        msg = "cannot subtract DatetimeArray from"
+        msg = "cannot subtract a datelike from a TimedeltaArray"
         with pytest.raises(TypeError, match=msg):
             tdi.values - dti
 
@@ -2172,13 +2172,7 @@ def test_dti_isub_tdi(self, tz_naive_fixture):
         result -= tdi.values
         tm.assert_index_equal(result, expected)
 
-        msg = "|".join(
-            [
-                "cannot perform __neg__ with this index type:",
-                "ufunc subtract cannot use operands with types",
-                "cannot subtract DatetimeArray from",
-            ]
-        )
+        msg = "cannot subtract a datelike from a TimedeltaArray"
         with pytest.raises(TypeError, match=msg):
             tdi.values -= dti
 
diff --git a/pandas/tests/arithmetic/test_period.py b/pandas/tests/arithmetic/test_period.py
index 5f93442cae4f6..7d215c940c031 100644
--- a/pandas/tests/arithmetic/test_period.py
+++ b/pandas/tests/arithmetic/test_period.py
@@ -753,7 +753,7 @@ def test_pi_add_sub_td64_array_non_tick_raises(self):
 
         with pytest.raises(TypeError, match=msg):
             rng - tdarr
-        msg = r"cannot subtract PeriodArray from timedelta64\[ns\]"
+        msg = r"cannot subtract period\[Q-DEC\]-dtype from TimedeltaArray"
         with pytest.raises(TypeError, match=msg):
             tdarr - rng