One thing I remember was if/when a couple of guys got into any altercation that culminated in a challenge to fight, one of them would say the term, "I call you down", which was the challenge to meet at an agreed upon location usually away from the school grounds away from teachers or other adults. Generally about 25 - 50 students would show up as bystanders and would form a circle around the two during the fight to cheer on their friend or who they wanted to win.
While I was at Gianini these after-school fights would most often take place along Sunset Blvd, a couple blocks from the school. Still the crowd forming the "circle" would draw the attention of adults (whether or not teachers or administrators) who would break-up the fights.
I don't recall these being referred to as a showdown, so I don't know if this was what you were looking for, but it is as close as I could come.