diff --git a/CHANGELOG.rst b/CHANGELOG.rst
index 17ac85d5..b2fc2dd3 100644
--- a/CHANGELOG.rst
+++ b/CHANGELOG.rst
@@ -6,6 +6,7 @@ Changelog
 
 * Enforce that the entire marker string is parsed (:issue:`687`)
 * Requirement parsing no longer automatically validates the URL (:issue:`120`)
+* Canonicalize names for requirements comparison (:issue:`644`)
 
 23.1 - 2023-04-12
 ~~~~~~~~~~~~~~~~~
diff --git a/src/packaging/requirements.py b/src/packaging/requirements.py
index e828c61f..0c00eba3 100644
--- a/src/packaging/requirements.py
+++ b/src/packaging/requirements.py
@@ -2,12 +2,13 @@
 # 2.0, and the BSD License. See the LICENSE file in the root of this repository
 # for complete details.
 
-from typing import Any, List, Optional, Set
+from typing import Any, Iterator, Optional, Set
 
 from ._parser import parse_requirement as _parse_requirement
 from ._tokenizer import ParserSyntaxError
 from .markers import Marker, _normalize_extra_values
 from .specifiers import SpecifierSet
+from .utils import canonicalize_name
 
 
 class InvalidRequirement(ValueError):
@@ -44,38 +45,44 @@ def __init__(self, requirement_string: str) -> None:
             self.marker = Marker.__new__(Marker)
             self.marker._markers = _normalize_extra_values(parsed.marker)
 
-    def __str__(self) -> str:
-        parts: List[str] = [self.name]
+    def _iter_parts(self, name: str) -> Iterator[str]:
+        yield name
 
         if self.extras:
             formatted_extras = ",".join(sorted(self.extras))
-            parts.append(f"[{formatted_extras}]")
+            yield f"[{formatted_extras}]"
 
         if self.specifier:
-            parts.append(str(self.specifier))
+            yield str(self.specifier)
 
         if self.url:
-            parts.append(f"@ {self.url}")
+            yield f"@ {self.url}"
             if self.marker:
-                parts.append(" ")
+                yield " "
 
         if self.marker:
-            parts.append(f"; {self.marker}")
+            yield f"; {self.marker}"
 
-        return "".join(parts)
+    def __str__(self) -> str:
+        return "".join(self._iter_parts(self.name))
 
     def __repr__(self) -> str:
         return f"<Requirement('{self}')>"
 
     def __hash__(self) -> int:
-        return hash((self.__class__.__name__, str(self)))
+        return hash(
+            (
+                self.__class__.__name__,
+                *self._iter_parts(canonicalize_name(self.name)),
+            )
+        )
 
     def __eq__(self, other: Any) -> bool:
         if not isinstance(other, Requirement):
             return NotImplemented
 
         return (
-            self.name == other.name
+            canonicalize_name(self.name) == canonicalize_name(other.name)
             and self.extras == other.extras
             and self.specifier == other.specifier
             and self.url == other.url
diff --git a/tests/test_requirements.py b/tests/test_requirements.py
index 45d3937e..491b3e03 100644
--- a/tests/test_requirements.py
+++ b/tests/test_requirements.py
@@ -26,6 +26,10 @@
     ),
 ]
 
+EQUIVALENT_DEPENDENCIES = [
+    ("scikit-learn==1.0.1", "scikit_learn==1.0.1"),
+]
+
 DIFFERENT_DEPENDENCIES = [
     ("package_one", "package_two"),
     ("packaging>20.1", "packaging>=20.1"),
@@ -632,12 +636,25 @@ def test_str_and_repr(
 
     @pytest.mark.parametrize("dep1, dep2", EQUAL_DEPENDENCIES)
     def test_equal_reqs_equal_hashes(self, dep1: str, dep2: str) -> None:
-        """Requirement objects created from equivalent strings should be equal."""
+        """Requirement objects created from equal strings should be equal."""
+        # GIVEN / WHEN
+        req1, req2 = Requirement(dep1), Requirement(dep2)
+
+        assert req1 == req2
+        assert hash(req1) == hash(req2)
+
+    @pytest.mark.parametrize("dep1, dep2", EQUIVALENT_DEPENDENCIES)
+    def test_equivalent_reqs_equal_hashes_unequal_strings(
+        self, dep1: str, dep2: str
+    ) -> None:
+        """Requirement objects created from equivalent strings should be equal,
+        even though their string representation will not."""
         # GIVEN / WHEN
         req1, req2 = Requirement(dep1), Requirement(dep2)
 
         assert req1 == req2
         assert hash(req1) == hash(req2)
+        assert str(req1) != str(req2)
 
     @pytest.mark.parametrize("dep1, dep2", DIFFERENT_DEPENDENCIES)
     def test_different_reqs_different_hashes(self, dep1: str, dep2: str) -> None: