diff --git a/src/packaging/_structures.py b/src/packaging/_structures.py
index 225e2eee..74ca8f0c 100644
--- a/src/packaging/_structures.py
+++ b/src/packaging/_structures.py
@@ -19,13 +19,13 @@ def __lt__(self, other: object) -> bool:
         return False
 
     def __le__(self, other: object) -> bool:
-        return False
+        return other is self
 
     def __eq__(self, other: object) -> bool:
-        return isinstance(other, self.__class__)
+        return other is self
 
     def __gt__(self, other: object) -> bool:
-        return True
+        return other is not self
 
     def __ge__(self, other: object) -> bool:
         return True
@@ -48,19 +48,19 @@ def __hash__(self) -> int:
         return hash(repr(self))
 
     def __lt__(self, other: object) -> bool:
-        return True
+        return other is not self
 
     def __le__(self, other: object) -> bool:
         return True
 
     def __eq__(self, other: object) -> bool:
-        return isinstance(other, self.__class__)
+        return other is self
 
     def __gt__(self, other: object) -> bool:
         return False
 
     def __ge__(self, other: object) -> bool:
-        return False
+        return other is self
 
     def __neg__(self: object) -> InfinityType:
         return Infinity
diff --git a/tests/test_structures.py b/tests/test_structures.py
index d85c9621..6b67d189 100644
--- a/tests/test_structures.py
+++ b/tests/test_structures.py
@@ -49,10 +49,24 @@ def test_infinity_equal() -> None:
     assert Infinity == Infinity
 
 
+def test_infinity_self_comparison() -> None:
+    assert not Infinity < Infinity
+    assert Infinity <= Infinity
+    assert not Infinity > Infinity
+    assert Infinity >= Infinity
+
+
 def test_negative_infinity_equal() -> None:
     assert NegativeInfinity == NegativeInfinity
 
 
+def test_negative_infinity_self_comparison() -> None:
+    assert not NegativeInfinity < NegativeInfinity
+    assert NegativeInfinity <= NegativeInfinity
+    assert not NegativeInfinity > NegativeInfinity
+    assert NegativeInfinity >= NegativeInfinity
+
+
 def test_negate_infinity() -> None:
     assert isinstance(-Infinity, NegativeInfinity.__class__)