The southpaw jab vs the orthodox jab is a very large part of fighting between southpaw/orthodox but I think ultimately southpaw/orthodox is won and lost on the Southpaw left straight and the Orthodox straight right. By that I mean everything revolves around the control of this. Either in the form of stopping/avoiding it or setting up counters off of it. Its kind of at the center of southpaw/orthodox and I think the fight is won and lost on how well a fighter handles it.

For example the natural counter for the southpaw jab is the straight right and vice versa either to the body or head (interestingly the southpaw is the one at a disadvantage because his liver is in front of his body, underneath his right arm, and therefore underneath his jab). But anyway this counter is only viable if your opponent is committing enough into his jab to allow you to counter it. So one part of the "jab war" between orthodox and southpaw is in seeing who can counter who's jab with this punch. If a fighter, in trying to win the jab war, commits too heavily into his jab, he puts himself in danger of walking face first into this counter.

Also the two other most effective counters when fighting with a southpaw (speaking as an orthodox fighter) is the right uppercut after slipping under their straight left and the left hook after blocking the straight left on your right arm. These are deadly counters.

Anyway my point is that in southpaw/orthodox everything ultimately comes back to how the left or straight right is handled and that plans should be developed around how to land them and how counter them. The jab is ultimately a means to that end.