diff --git a/src/hyperlink/_url.py b/src/hyperlink/_url.py
index 8797b5c..308d3e6 100644
--- a/src/hyperlink/_url.py
+++ b/src/hyperlink/_url.py
@@ -1259,11 +1259,10 @@ def __hash__(self):
         # type: () -> int
         return hash(
             (
-                self.__class__,
                 self.scheme,
                 self.userinfo,
                 self.host,
-                self.path,
+                () if self.path in _ROOT_PATHS else self.path,
                 self.query,
                 self.fragment,
                 self.port,
@@ -2379,20 +2378,7 @@ def __ne__(self, other):
 
     def __hash__(self):
         # type: () -> int
-        return hash(
-            (
-                self.__class__,
-                self.scheme,
-                self.userinfo,
-                self.host,
-                self.path,
-                self.query,
-                self.fragment,
-                self.port,
-                self.rooted,
-                self.uses_netloc,
-            )
-        )
+        return hash(self.normalize().to_uri())
 
     # # Begin Twisted Compat Code
     asURI = to_uri
diff --git a/src/hyperlink/test/test_decoded_url.py b/src/hyperlink/test/test_decoded_url.py
index 4845257..202c029 100644
--- a/src/hyperlink/test/test_decoded_url.py
+++ b/src/hyperlink/test/test_decoded_url.py
@@ -172,6 +172,33 @@ def test_equality_and_hashability(self):
 
         assert len(durl_map) == 3
 
+    def test_hash_inherited_equality(self):
+        # type: () -> None
+        class InheritedURL(DecodedURL):
+            pass
+
+        url = DecodedURL.from_text("http://example.com/a")
+        inherited = InheritedURL.from_text("http://example.com/a")
+        assert url == inherited
+        assert hash(url) == hash(inherited)
+        assert {url: "value"}[inherited] == "value"
+
+    def test_hash_equivalent_normalized_urls(self):
+        # type: () -> None
+        pairs = [
+            ("http://example.com", "http://example.com/"),
+            ("http://example.com/a", "HTTP://EXAMPLE.COM/a"),
+            ("http://example.com/a", "http://example.com/b/../a"),
+            ("http://example.com/~", "http://example.com/%7E"),
+        ]
+        for left, right in pairs:
+            left_url = DecodedURL.from_text(left)
+            right_url = DecodedURL.from_text(right)
+            assert left_url == right_url
+            assert hash(left_url) == hash(right_url)
+            assert len({left_url, right_url}) == 1
+            assert {left_url: "value"}[right_url] == "value"
+
     def test_replace_roundtrip(self):
         # type: () -> None
         durl = DecodedURL.from_text(TOTAL_URL)
diff --git a/src/hyperlink/test/test_url.py b/src/hyperlink/test/test_url.py
index 37c9172..bfbfc29 100644
--- a/src/hyperlink/test/test_url.py
+++ b/src/hyperlink/test/test_url.py
@@ -1295,6 +1295,26 @@ def test_hash(self):
 
         assert hash(URL()) == hash(URL())  # slightly more sanity
 
+    def test_hash_inherited_equality(self):
+        # type: () -> None
+        class InheritedURL(URL):
+            pass
+
+        url = URL.from_text("http://example.com/a")
+        inherited = InheritedURL.from_text("http://example.com/a")
+        assert url == inherited
+        assert hash(url) == hash(inherited)
+        assert {url: "value"}[inherited] == "value"
+
+    def test_hash_equivalent_root_paths(self):
+        # type: () -> None
+        without_slash = URL.from_text("http://example.com")
+        with_slash = URL.from_text("http://example.com/")
+        assert without_slash == with_slash
+        assert hash(without_slash) == hash(with_slash)
+        assert len({without_slash, with_slash}) == 1
+        assert {without_slash: "value"}[with_slash] == "value"
+
     def test_dir(self):
         # type: () -> None
         url = URL()