A bike moving forward will self correct to stay up until it loses speed. A bike moving backward will quickly go into a hard turn right or left and fall over. Gyroscopic moments for wheels in both scenarios are the same. The difference is that as a forward-moving bike falls to one side, the front wheel straightens out underneath it. As a backward moving bike (or any bike with a turning rear wheel), starts to fall, the front (now rear) wheel turns further to the side as it falls rather than straightening out, and the bike goes into a very sharp turn and falls.
As an addendum, that's why recumbent bikes stick with a rear drive and front steer, which requires very long chains, rather than driving the front and steering with the rear (which at first blush seems like a much simpler configuration - the pedals are up near the front wheel, and the hands are near the rear).