diff --git a/tornado/httputil.py b/tornado/httputil.py
index 4bd17786d..e03ba61a5 100644
--- a/tornado/httputil.py
+++ b/tornado/httputil.py
@@ -874,6 +874,8 @@ def _parse_request_range(
     (None, 0)
     >>> _parse_request_range("bytes=")
     (None, None)
+    >>> _parse_request_range("BYTES=1-2")
+    (1, 3)
     >>> _parse_request_range("foo=42")
     >>> _parse_request_range("bytes=1-2,6-10")
 
@@ -885,7 +887,7 @@ def _parse_request_range(
     """
     unit, _, value = range_header.partition("=")
     unit, value = unit.strip(), value.strip()
-    if unit != "bytes":
+    if unit.lower() != "bytes":
         return None
     start_b, _, end_b = value.partition("-")
     try:
diff --git a/tornado/test/httputil_test.py b/tornado/test/httputil_test.py
index 4e966eb50..5619ef286 100644
--- a/tornado/test/httputil_test.py
+++ b/tornado/test/httputil_test.py
@@ -14,6 +14,7 @@
     HTTPServerRequest,
     ParseMultipartConfig,
     RequestStartLine,
+    _parse_request_range,
     format_timestamp,
     parse_cookie,
     parse_multipart_form_data,
@@ -623,6 +624,31 @@ def test_parse_request_start_line(self):
         self.assertEqual(parsed_start_line.version, self.VERSION)
 
 
+class ParseRequestRangeTest(unittest.TestCase):
+    """Tests for httputil._parse_request_range."""
+
+    def test_lowercase_unit(self):
+        self.assertEqual(_parse_request_range("bytes=1-2"), (1, 3))
+
+    def test_uppercase_unit_accepted(self):
+        # Per RFC 7233 section 2.1: "all rules derived from token are to be
+        # compared case-insensitively, like range-unit and acceptable-ranges."
+        # Pre-fix, an uppercase "BYTES" unit returned None.
+        self.assertEqual(_parse_request_range("BYTES=1-2"), (1, 3))
+
+    def test_mixed_case_unit_accepted(self):
+        self.assertEqual(_parse_request_range("Bytes=1-2"), (1, 3))
+        self.assertEqual(_parse_request_range("bYtEs=1-2"), (1, 3))
+
+    def test_uppercase_unit_with_suffix_range(self):
+        self.assertEqual(_parse_request_range("BYTES=6-"), (6, None))
+        self.assertEqual(_parse_request_range("BYTES=-6"), (-6, None))
+
+    def test_non_bytes_unit_still_rejected(self):
+        # An unknown unit is rejected regardless of case.
+        self.assertIsNone(_parse_request_range("items=1-2"))
+
+
 class ParseCookieTest(unittest.TestCase):
     # These tests copied from Django:
     # https://github.com/django/django/pull/6277/commits/da810901ada1cae9fc1f018f879f11a7fb467b28