diff --git a/scrapy/downloadermiddlewares/redirect.py b/scrapy/downloadermiddlewares/redirect.py
index a26715e12..36bd9a849 100644
--- a/scrapy/downloadermiddlewares/redirect.py
+++ b/scrapy/downloadermiddlewares/redirect.py
@@ -38,18 +38,19 @@ def _build_redirect_request(source_request, *, url, **kwargs):
             or default_ports.get(parsed_redirect_request.scheme),
         )
 
-        if redirect_scheme != "https" or source_host != redirect_host:
-            if has_cookie_header:
-                del redirect_request.headers["Cookie"]
+        if has_cookie_header and (
+            (source_scheme != redirect_scheme and redirect_scheme != "https")
+            or source_host != redirect_host
+        ):
+            del redirect_request.headers["Cookie"]
 
-        if (
+        # https://fetch.spec.whatwg.org/#ref-for-cors-non-wildcard-request-header-name
+        if has_authorization_header and (
             source_scheme != redirect_scheme
             or source_host != redirect_host
             or source_port != redirect_port
         ):
-            # https://fetch.spec.whatwg.org/#ref-for-cors-non-wildcard-request-header-name
-            if has_authorization_header:
-                del redirect_request.headers["Authorization"]
+            del redirect_request.headers["Authorization"]
 
     return redirect_request
 
diff --git a/tests/test_downloadermiddleware_redirect.py b/tests/test_downloadermiddleware_redirect.py
index c14f87d91..3b0b910ee 100644
--- a/tests/test_downloadermiddleware_redirect.py
+++ b/tests/test_downloadermiddleware_redirect.py
@@ -270,6 +270,23 @@ class RedirectMiddlewareTest(unittest.TestCase):
         self.assertIsInstance(internal_redirect_request, Request)
         self.assertEqual(original_request.headers, internal_redirect_request.headers)
 
+        # Redirects to the same origin (same scheme, same domain, same port)
+        # keep all headers also when the scheme is http.
+        http_request = Request(
+            "http://example.com",
+            headers={**safe_headers, **cookie_header, **authorization_header},
+        )
+        http_response = Response(
+            "http://example.com",
+            headers={"Location": "http://example.com/a"},
+            status=301,
+        )
+        http_redirect_request = self.mw.process_response(
+            http_request, http_response, self.spider
+        )
+        self.assertIsInstance(http_redirect_request, Request)
+        self.assertEqual(http_request.headers, http_redirect_request.headers)
+
         # For default ports, whether the port is explicit or implicit does not
         # affect the outcome, it is still the same origin.
         to_explicit_port_response = Response(
@@ -334,10 +351,6 @@ class RedirectMiddlewareTest(unittest.TestCase):
         # A scheme upgrade (http → https) drops the Authorization header
         # because the origin changes, but keeps the Cookie header because the
         # domain remains the same.
-        http_request = Request(
-            "http://example.com",
-            headers={**safe_headers, **cookie_header, **authorization_header},
-        )
         upgrade_response = Response(
             "http://example.com",
             headers={"Location": "https://example.com/a"},