diff --git a/pandas/core/arrays/interval.py b/pandas/core/arrays/interval.py
index 161cf3bf3a677..f8ece2a9fe7d4 100644
--- a/pandas/core/arrays/interval.py
+++ b/pandas/core/arrays/interval.py
@@ -1,3 +1,4 @@
+import operator
 from operator import le, lt
 import textwrap
 from typing import TYPE_CHECKING, Optional, Tuple, Union, cast
@@ -12,6 +13,7 @@
     IntervalMixin,
     intervals_to_interval_bounds,
 )
+from pandas._libs.missing import NA
 from pandas._typing import ArrayLike, Dtype
 from pandas.compat.numpy import function as nv
 from pandas.util._decorators import Appender
@@ -48,7 +50,7 @@
 from pandas.core.construction import array, extract_array
 from pandas.core.indexers import check_array_indexer
 from pandas.core.indexes.base import ensure_index
-from pandas.core.ops import unpack_zerodim_and_defer
+from pandas.core.ops import invalid_comparison, unpack_zerodim_and_defer
 
 if TYPE_CHECKING:
     from pandas import Index
@@ -520,8 +522,7 @@ def __setitem__(self, key, value):
         self._left[key] = value_left
         self._right[key] = value_right
 
-    @unpack_zerodim_and_defer("__eq__")
-    def __eq__(self, other):
+    def _cmp_method(self, other, op):
         # ensure pandas array for list-like and eliminate non-interval scalars
         if is_list_like(other):
             if len(self) != len(other):
@@ -529,7 +530,7 @@ def __eq__(self, other):
             other = array(other)
         elif not isinstance(other, Interval):
             # non-interval scalar -> no matches
-            return np.zeros(len(self), dtype=bool)
+            return invalid_comparison(self, other, op)
 
         # determine the dtype of the elements we want to compare
         if isinstance(other, Interval):
@@ -543,7 +544,8 @@ def __eq__(self, other):
             # extract intervals if we have interval categories with matching closed
             if is_interval_dtype(other_dtype):
                 if self.closed != other.categories.closed:
-                    return np.zeros(len(self), dtype=bool)
+                    return invalid_comparison(self, other, op)
+
                 other = other.categories.take(
                     other.codes, allow_fill=True, fill_value=other.categories._na_value
                 )
@@ -551,27 +553,70 @@ def __eq__(self, other):
         # interval-like -> need same closed and matching endpoints
         if is_interval_dtype(other_dtype):
             if self.closed != other.closed:
-                return np.zeros(len(self), dtype=bool)
-            return (self._left == other.left) & (self._right == other.right)
+                return invalid_comparison(self, other, op)
+            elif not isinstance(other, Interval):
+                other = type(self)(other)
+
+            if op is operator.eq:
+                return (self._left == other.left) & (self._right == other.right)
+            elif op is operator.ne:
+                return (self._left != other.left) | (self._right != other.right)
+            elif op is operator.gt:
+                return (self._left > other.left) | (
+                    (self._left == other.left) & (self._right > other.right)
+                )
+            elif op is operator.ge:
+                return (self == other) | (self > other)
+            elif op is operator.lt:
+                return (self._left < other.left) | (
+                    (self._left == other.left) & (self._right < other.right)
+                )
+            else:
+                # operator.lt
+                return (self == other) | (self < other)
 
         # non-interval/non-object dtype -> no matches
         if not is_object_dtype(other_dtype):
-            return np.zeros(len(self), dtype=bool)
+            return invalid_comparison(self, other, op)
 
         # object dtype -> iteratively check for intervals
         result = np.zeros(len(self), dtype=bool)
         for i, obj in enumerate(other):
-            # need object to be an Interval with same closed and endpoints
-            if (
-                isinstance(obj, Interval)
-                and self.closed == obj.closed
-                and self._left[i] == obj.left
-                and self._right[i] == obj.right
-            ):
-                result[i] = True
-
+            try:
+                result[i] = op(self[i], obj)
+            except TypeError:
+                if obj is NA:
+                    # comparison with np.nan returns NA
+                    # github.com/pandas-dev/pandas/pull/37124#discussion_r509095092
+                    result[i] = op is operator.ne
+                else:
+                    raise
         return result
 
+    @unpack_zerodim_and_defer("__eq__")
+    def __eq__(self, other):
+        return self._cmp_method(other, operator.eq)
+
+    @unpack_zerodim_and_defer("__ne__")
+    def __ne__(self, other):
+        return self._cmp_method(other, operator.ne)
+
+    @unpack_zerodim_and_defer("__gt__")
+    def __gt__(self, other):
+        return self._cmp_method(other, operator.gt)
+
+    @unpack_zerodim_and_defer("__ge__")
+    def __ge__(self, other):
+        return self._cmp_method(other, operator.ge)
+
+    @unpack_zerodim_and_defer("__lt__")
+    def __lt__(self, other):
+        return self._cmp_method(other, operator.lt)
+
+    @unpack_zerodim_and_defer("__le__")
+    def __le__(self, other):
+        return self._cmp_method(other, operator.le)
+
     def fillna(self, value=None, method=None, limit=None):
         """
         Fill NA/NaN values using the specified method.
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index 1bd71f00b534d..2061e652a4c01 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -1074,19 +1074,6 @@ def _is_all_dates(self) -> bool:
 
     # TODO: arithmetic operations
 
-    # GH#30817 until IntervalArray implements inequalities, get them from Index
-    def __lt__(self, other):
-        return Index.__lt__(self, other)
-
-    def __le__(self, other):
-        return Index.__le__(self, other)
-
-    def __gt__(self, other):
-        return Index.__gt__(self, other)
-
-    def __ge__(self, other):
-        return Index.__ge__(self, other)
-
 
 def _is_valid_endpoint(endpoint) -> bool:
     """
diff --git a/pandas/tests/extension/base/methods.py b/pandas/tests/extension/base/methods.py
index e973b1247941f..29a59cdefbd83 100644
--- a/pandas/tests/extension/base/methods.py
+++ b/pandas/tests/extension/base/methods.py
@@ -447,7 +447,7 @@ def test_repeat(self, data, repeats, as_series, use_numpy):
     @pytest.mark.parametrize(
         "repeats, kwargs, error, msg",
         [
-            (2, dict(axis=1), ValueError, "'axis"),
+            (2, dict(axis=1), ValueError, "axis"),
             (-1, dict(), ValueError, "negative"),
             ([1, 2], dict(), ValueError, "shape"),
             (2, dict(foo="bar"), TypeError, "'foo'"),
diff --git a/pandas/tests/indexes/interval/test_interval.py b/pandas/tests/indexes/interval/test_interval.py
index 67e031b53e44e..157446b1fff5d 100644
--- a/pandas/tests/indexes/interval/test_interval.py
+++ b/pandas/tests/indexes/interval/test_interval.py
@@ -579,9 +579,11 @@ def test_comparison(self):
         actual = self.index == self.index.left
         tm.assert_numpy_array_equal(actual, np.array([False, False]))
 
-        msg = (
-            "not supported between instances of 'int' and "
-            "'pandas._libs.interval.Interval'"
+        msg = "|".join(
+            [
+                "not supported between instances of 'int' and '.*.Interval'",
+                r"Invalid comparison between dtype=interval\[int64\] and ",
+            ]
         )
         with pytest.raises(TypeError, match=msg):
             self.index > 0