diff --git a/mysql-test/include/mix1.inc b/mysql-test/include/mix1.inc
index a55e0f22a7c..e2a22303d9d 100644
--- a/mysql-test/include/mix1.inc
+++ b/mysql-test/include/mix1.inc
@@ -996,6 +996,29 @@ SELECT c,b   FROM t1 GROUP BY c,b;
 
 DROP TABLE t1;
 
+#
+# Bug #31001: ORDER BY DESC in InnoDB not working
+#
+CREATE TABLE t1 (a INT, b INT, PRIMARY KEY (a), INDEX b (b)) ENGINE=InnoDB;
+INSERT INTO t1(a,b) VALUES (1,1), (2,2), (3,2);
+
+#The two queries below should produce different results, but they don't.
+query_vertical EXPLAIN SELECT * FROM t1 WHERE b=2 ORDER BY a ASC;
+SELECT * FROM t1 WHERE b=2 ORDER BY a ASC;
+query_vertical EXPLAIN SELECT * FROM t1 WHERE b=2 ORDER BY a DESC;
+SELECT * FROM t1 WHERE b=2 ORDER BY a DESC;
+
+query_vertical EXPLAIN SELECT * FROM t1 ORDER BY b ASC, a ASC;
+SELECT * FROM t1 ORDER BY b ASC, a ASC;
+query_vertical EXPLAIN SELECT * FROM t1 ORDER BY b DESC, a DESC;
+SELECT * FROM t1 ORDER BY b DESC, a DESC;
+query_vertical EXPLAIN SELECT * FROM t1 ORDER BY b ASC, a DESC;
+SELECT * FROM t1 ORDER BY b ASC, a DESC;
+query_vertical EXPLAIN SELECT * FROM t1 ORDER BY b DESC, a ASC;
+SELECT * FROM t1 ORDER BY b DESC, a ASC;
+
+DROP TABLE t1;
+
 --echo End of 5.0 tests
 
 # Fix for BUG#19243 "wrong LAST_INSERT_ID() after ON DUPLICATE KEY
diff --git a/mysql-test/r/innodb_mysql.result b/mysql-test/r/innodb_mysql.result
index 3293f05a1f9..0757936a8f7 100644
--- a/mysql-test/r/innodb_mysql.result
+++ b/mysql-test/r/innodb_mysql.result
@@ -1190,6 +1190,103 @@ c	b
 3	1
 3	2
 DROP TABLE t1;
+CREATE TABLE t1 (a INT, b INT, PRIMARY KEY (a), INDEX b (b)) ENGINE=InnoDB;
+INSERT INTO t1(a,b) VALUES (1,1), (2,2), (3,2);
+EXPLAIN SELECT * FROM t1 WHERE b=2 ORDER BY a ASC;
+id	1
+select_type	SIMPLE
+table	t1
+type	ref
+possible_keys	b
+key	b
+key_len	5
+ref	const
+rows	1
+Extra	Using where; Using index
+SELECT * FROM t1 WHERE b=2 ORDER BY a ASC;
+a	b
+2	2
+3	2
+EXPLAIN SELECT * FROM t1 WHERE b=2 ORDER BY a DESC;
+id	1
+select_type	SIMPLE
+table	t1
+type	ref
+possible_keys	b
+key	b
+key_len	5
+ref	const
+rows	1
+Extra	Using where; Using index
+SELECT * FROM t1 WHERE b=2 ORDER BY a DESC;
+a	b
+3	2
+2	2
+EXPLAIN SELECT * FROM t1 ORDER BY b ASC, a ASC;
+id	1
+select_type	SIMPLE
+table	t1
+type	index
+possible_keys	NULL
+key	b
+key_len	5
+ref	NULL
+rows	3
+Extra	Using index
+SELECT * FROM t1 ORDER BY b ASC, a ASC;
+a	b
+1	1
+2	2
+3	2
+EXPLAIN SELECT * FROM t1 ORDER BY b DESC, a DESC;
+id	1
+select_type	SIMPLE
+table	t1
+type	index
+possible_keys	NULL
+key	b
+key_len	5
+ref	NULL
+rows	3
+Extra	Using index
+SELECT * FROM t1 ORDER BY b DESC, a DESC;
+a	b
+3	2
+2	2
+1	1
+EXPLAIN SELECT * FROM t1 ORDER BY b ASC, a DESC;
+id	1
+select_type	SIMPLE
+table	t1
+type	index
+possible_keys	NULL
+key	b
+key_len	5
+ref	NULL
+rows	3
+Extra	Using index; Using filesort
+SELECT * FROM t1 ORDER BY b ASC, a DESC;
+a	b
+1	1
+3	2
+2	2
+EXPLAIN SELECT * FROM t1 ORDER BY b DESC, a ASC;
+id	1
+select_type	SIMPLE
+table	t1
+type	index
+possible_keys	NULL
+key	b
+key_len	5
+ref	NULL
+rows	3
+Extra	Using index; Using filesort
+SELECT * FROM t1 ORDER BY b DESC, a ASC;
+a	b
+2	2
+3	2
+1	1
+DROP TABLE t1;
 End of 5.0 tests
 CREATE TABLE `t2` (
 `k` int(11) NOT NULL auto_increment,
diff --git a/sql/sql_select.cc b/sql/sql_select.cc
index b903e7ae13c..c6f114a1165 100644
--- a/sql/sql_select.cc
+++ b/sql/sql_select.cc
@@ -12353,6 +12353,12 @@ static int test_if_order_by_key(ORDER *order, TABLE *table, uint idx,
 
         for (; const_key_parts & 1 ; const_key_parts>>= 1)
           key_part++; 
+        /*
+         The primary and secondary key parts were all const (i.e. there's
+         one row).  The sorting doesn't matter.
+        */
+        if (key_part == key_part_end && reverse == 0)
+          DBUG_RETURN(1);
       }
       else
         DBUG_RETURN(0);
@@ -12995,7 +13001,7 @@ check_reverse_order:
 	select->quick=tmp;
       }
     }
-    else if (tab->ref.key >= 0 && tab->ref.key_parts < used_key_parts)
+    else if (tab->ref.key >= 0 && tab->ref.key_parts <= used_key_parts)
     {
       /*
 	SELECT * FROM t1 WHERE a=1 ORDER BY a DESC,b DESC