diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index b143ff0aa9c02..2f2fc453f02b8 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -80,16 +80,7 @@ def wrapper(left, right):
     cache=True,
 )
 @inherit_names(
-    [
-        "__iter__",
-        "mean",
-        "freq",
-        "freqstr",
-        "_ndarray_values",
-        "asi8",
-        "_box_values",
-        "_box_func",
-    ],
+    ["mean", "freq", "freqstr", "asi8", "_box_values", "_box_func"],
     DatetimeLikeArrayMixin,
 )
 class DatetimeIndexOpsMixin(ExtensionIndex):
diff --git a/pandas/core/indexes/extension.py b/pandas/core/indexes/extension.py
index 66b551f654bf1..04b4b275bf90a 100644
--- a/pandas/core/indexes/extension.py
+++ b/pandas/core/indexes/extension.py
@@ -196,6 +196,9 @@ class ExtensionIndex(Index):
     Index subclass for indexes backed by ExtensionArray.
     """
 
+    # The base class already passes through to _data:
+    #  size, __len__, dtype
+
     _data: ExtensionArray
 
     __eq__ = _make_wrapped_comparison_op("__eq__")
@@ -205,6 +208,9 @@ class ExtensionIndex(Index):
     __le__ = _make_wrapped_comparison_op("__le__")
     __ge__ = _make_wrapped_comparison_op("__ge__")
 
+    # ---------------------------------------------------------------------
+    # NDarray-Like Methods
+
     def __getitem__(self, key):
         result = self._data[key]
         if isinstance(result, type(self._data)):
@@ -217,6 +223,8 @@ def __getitem__(self, key):
     def __iter__(self):
         return self._data.__iter__()
 
+    # ---------------------------------------------------------------------
+
     @property
     def _ndarray_values(self) -> np.ndarray:
         return self._data._ndarray_values
@@ -235,6 +243,10 @@ def repeat(self, repeats, axis=None):
         result = self._data.repeat(repeats, axis=axis)
         return self._shallow_copy(result)
 
+    def _concat_same_dtype(self, to_concat, name):
+        arr = type(self._data)._concat_same_type(to_concat)
+        return type(self)._simple_new(arr, name=name)
+
     @Appender(Index.take.__doc__)
     def take(self, indices, axis=0, allow_fill=True, fill_value=None, **kwargs):
         nv.validate_take(tuple(), kwargs)
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index 03fb8db2e1e1e..f77785a29b262 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -183,23 +183,10 @@ def func(intvidx_self, other, sort=False):
 )
 @inherit_names(["set_closed", "to_tuples"], IntervalArray, wrap=True)
 @inherit_names(
-    [
-        "__len__",
-        "__array__",
-        "overlaps",
-        "contains",
-        "size",
-        "dtype",
-        "left",
-        "right",
-        "length",
-    ],
-    IntervalArray,
+    ["__array__", "overlaps", "contains", "left", "right", "length"], IntervalArray,
 )
 @inherit_names(
-    ["is_non_overlapping_monotonic", "mid", "_ndarray_values", "closed"],
-    IntervalArray,
-    cache=True,
+    ["is_non_overlapping_monotonic", "mid", "closed"], IntervalArray, cache=True,
 )
 class IntervalIndex(IntervalMixin, ExtensionIndex):
     _typ = "intervalindex"
@@ -943,18 +930,6 @@ def insert(self, loc, item):
         new_right = self.right.insert(loc, right_insert)
         return self._shallow_copy(new_left, new_right)
 
-    def _concat_same_dtype(self, to_concat, name):
-        """
-        assert that we all have the same .closed
-        we allow a 0-len index here as well
-        """
-        if not len({i.closed for i in to_concat if len(i)}) == 1:
-            raise ValueError(
-                "can only append two IntervalIndex objects "
-                "that are closed on the same side"
-            )
-        return super()._concat_same_dtype(to_concat, name)
-
     @Appender(_index_shared_docs["take"] % _index_doc_kwargs)
     def take(self, indices, axis=0, allow_fill=True, fill_value=None, **kwargs):
         result = self._data.take(
@@ -962,14 +937,6 @@ def take(self, indices, axis=0, allow_fill=True, fill_value=None, **kwargs):
         )
         return self._shallow_copy(result)
 
-    def __getitem__(self, value):
-        result = self._data[value]
-        if isinstance(result, IntervalArray):
-            return self._shallow_copy(result)
-        else:
-            # scalar
-            return result
-
     # --------------------------------------------------------------------
     # Rendering Methods
     # __repr__ associated methods are based on MultiIndex
diff --git a/pandas/tests/indexes/interval/test_interval.py b/pandas/tests/indexes/interval/test_interval.py
index 47a0ba7fe0f21..d010060880703 100644
--- a/pandas/tests/indexes/interval/test_interval.py
+++ b/pandas/tests/indexes/interval/test_interval.py
@@ -673,10 +673,7 @@ def test_append(self, closed):
         )
         tm.assert_index_equal(result, expected)
 
-        msg = (
-            "can only append two IntervalIndex objects that are closed "
-            "on the same side"
-        )
+        msg = "Intervals must all be closed on the same side"
         for other_closed in {"left", "right", "both", "neither"} - {closed}:
             index_other_closed = IntervalIndex.from_arrays(
                 [0, 1], [1, 2], closed=other_closed