It's funny you should say that. You're not far off with that but you've got the institutions wrong. It wasn't between schools, (neither of which I attended, btw), but pubs. The Birley Gang vs the 'Arra.
(The Birley Hotel is at Frecheville and not to be confused with the Frecheville, which was at Birley!) You'll know the Old Harrow at Basegreen, Gleadless. (Fox Lane / White Lane.) Sadly it's now The Bay of Bengal eatery.
I grew up near the Birley Hotel but spent my youth in the Arra, somehow steering clear of all / any shit shows that occurred.