diff --git a/mysql-test/include/mix1.inc b/mysql-test/include/mix1.inc
index 32a3e6b3ebd..65c97db028a 100644
--- a/mysql-test/include/mix1.inc
+++ b/mysql-test/include/mix1.inc
@@ -939,6 +939,29 @@ alter table t1 add index(a(1024));
 show create table t1;
 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 26000fa27b7..99def85fe3b 100644
--- a/mysql-test/r/innodb_mysql.result
+++ b/mysql-test/r/innodb_mysql.result
@@ -1123,6 +1123,103 @@ t1	CREATE TABLE `t1` (
   KEY `a` (`a`(255))
 ) ENGINE=InnoDB DEFAULT CHARSET=utf8
 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 52a26df1088..a36cffcc2cb 100644
--- a/sql/sql_select.cc
+++ b/sql/sql_select.cc
@@ -12306,6 +12306,12 @@ 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);
@@ -12723,7 +12729,7 @@ test_if_skip_sort_order(JOIN_TAB *tab,ORDER *order,ha_rows select_limit,
 	  }
 	  DBUG_RETURN(1);
 	}
-	if (tab->ref.key_parts < used_key_parts)
+	if (tab->ref.key_parts <= used_key_parts)
 	{
 	  /*
 	    SELECT * FROM t1 WHERE a=1 ORDER BY a DESC,b DESC