Saint Louis University's basketball team has played 19 games this season and and has alteranted wins and losses all year also had a win and a loss to end last season, so for 21 straight games this has been this pattern:
Records on such a streak are hard to come by because, in my opinion, it's unusual.
The 1974 Phillies - Which started on June 3 with a win against Atlanta and continued through June 20 with a loss against the Mets. Then on June 21, the Phillies lost again to the Mets, breaking the spell. The 1974 Phillies finished 80-82. The streak started with six homes game, followed by six road games, and then four games at home.
The 1981 Dodgers - They lost at Chicago on June 6 and then won the next day against the Cubs. This continued through June 11, for a total of four games. Then there was a strike for nearly two months. On August 10, the season restarted and the Dodgers were in Cincinnati and won. The next day, they lost. This kept up through August 21 (win at St. Louis). On August 22, the Dodgers won again. The Dodgers finished 63-47 and ended up winning the World Series.
The 1994 Indianapolis Colts had a 12-game from Week 4 through Week 16 (with a bye in Week 11) before winning the last game of the year. The Colts were 8-8 that year.
For the more math-inclined, you can check out Alan Reifman's Hot Hand in Sports site to see how unusual a 21-game stretch of alternating Ws and Ls is. According to the linked calculator on the site, the chances of going WL to get to 21 games is 1 in 352,716. (If I copied it correctly.)
The Billikens play at Rhode Island on February 1. And they will likely be underdogs in that game. But on February 5, the Billikens host Xavier, whom they already upset on the road, so the magic of the alternating Ws and Ls may come to an end.