diff --git a/src/auditwheel/policy/__init__.py b/src/auditwheel/policy/__init__.py
index 20a72c12..7c9b1585 100644
--- a/src/auditwheel/policy/__init__.py
+++ b/src/auditwheel/policy/__init__.py
@@ -4,6 +4,7 @@
 import logging
 import platform as _platform_module
 import re
+import struct
 import sys
 from collections import defaultdict
 from os.path import abspath, dirname, join
@@ -21,9 +22,6 @@
 
 logger = logging.getLogger(__name__)
 
-# https://docs.python.org/3/library/platform.html#platform.architecture
-bits = 8 * (8 if sys.maxsize > 2**32 else 4)
-
 _POLICY_JSON_MAP = {
     Libc.GLIBC: _HERE / "manylinux-policy.json",
     Libc.MUSL: _HERE / "musllinux-policy.json",
@@ -224,10 +222,15 @@ def get_req_external(libs: set[str], whitelist: set[str]) -> set[str]:
         return ret
 
 
-def get_arch_name() -> str:
+def get_arch_name(*, bits: int | None = None) -> str:
     machine = _platform_module.machine()
     if sys.platform == "darwin" and machine == "arm64":
         return "aarch64"
+
+    if bits is None:
+        # c.f. https://github.com/pypa/packaging/pull/711
+        bits = 8 * struct.calcsize("P")
+
     if machine in {"x86_64", "i686"}:
         return {64: "x86_64", 32: "i686"}[bits]
     if machine in {"aarch64", "armv8l"}:
diff --git a/tests/unit/test_policy.py b/tests/unit/test_policy.py
index 9fd03c24..d05418f3 100644
--- a/tests/unit/test_policy.py
+++ b/tests/unit/test_policy.py
@@ -1,8 +1,10 @@
 from __future__ import annotations
 
+import platform
 import re
+import struct
+import sys
 from contextlib import nullcontext as does_not_raise
-from unittest.mock import patch
 
 import pytest
 
@@ -32,8 +34,6 @@ def raises(exception, match=None, escape=True):
     return pytest.raises(exception, match=match)
 
 
-@patch("auditwheel.policy._platform_module.machine")
-@patch("auditwheel.policy.bits", 32)
 @pytest.mark.parametrize(
     "reported_arch,expected_arch",
     [
@@ -45,14 +45,12 @@ def raises(exception, match=None, escape=True):
         ("x86_64", "i686"),
     ],
 )
-def test_32bits_arch_name(machine_mock, reported_arch, expected_arch):
-    machine_mock.return_value = reported_arch
-    machine = get_arch_name()
+def test_32bits_arch_name(reported_arch, expected_arch, monkeypatch):
+    monkeypatch.setattr(platform, "machine", lambda: reported_arch)
+    machine = get_arch_name(bits=32)
     assert machine == expected_arch
 
 
-@patch("auditwheel.policy._platform_module.machine")
-@patch("auditwheel.policy.bits", 64)
 @pytest.mark.parametrize(
     "reported_arch,expected_arch",
     [
@@ -63,12 +61,35 @@ def test_32bits_arch_name(machine_mock, reported_arch, expected_arch):
         ("x86_64", "x86_64"),
     ],
 )
-def test_64bits_arch_name(machine_mock, reported_arch, expected_arch):
-    machine_mock.return_value = reported_arch
-    machine = get_arch_name()
+def test_64bits_arch_name(reported_arch, expected_arch, monkeypatch):
+    monkeypatch.setattr(platform, "machine", lambda: reported_arch)
+    machine = get_arch_name(bits=64)
     assert machine == expected_arch
 
 
+@pytest.mark.parametrize(
+    "maxsize, sizeof_voidp, expected",
+    [
+        # 64-bit
+        (9223372036854775807, 8, "x86_64"),
+        # 32-bit
+        (2147483647, 4, "i686"),
+        # 64-bit w/ 32-bit sys.maxsize: GraalPy, IronPython, Jython
+        (2147483647, 8, "x86_64"),
+    ],
+)
+def test_arch_name_bits(maxsize, sizeof_voidp, expected, monkeypatch):
+    def _calcsize(fmt):
+        assert fmt == "P"
+        return sizeof_voidp
+
+    monkeypatch.setattr(platform, "machine", lambda: "x86_64")
+    monkeypatch.setattr(sys, "maxsize", maxsize)
+    monkeypatch.setattr(struct, "calcsize", _calcsize)
+    machine = get_arch_name()
+    assert machine == expected
+
+
 @pytest.mark.parametrize(
     "name,expected",
     [