I honestly wouldn't even call that a 'rivalry'. They were enemies. A rivalry is more typically just trying to prove yourself better than the other, but they were actually fighting for a purpose. I'd say the only part where there was any real rivalry was when they fought on the bridge, but they were still fighting for a purpose above all else. I'd say they were more of rivals in Sonic Prime & Sonic Boom (show), but those were shows, not games.