Disclaimer: I mentioned psychopaths and violent people, but that's in a context of an actual war and actual killing. If we only speak about "fighting" metaphorically, we need to appropriately redefine what it means to be "violent". In context of verbal internet wars, the analogy of psychopaths would be trolls, and the analogy of people who enjoy violence would be people who enjoy winning debates.

For the internet version of Genghis Khan, the greatest joy is to defeat his enemies in a public discourse, make them unpopular, destroy their websites, and take over their followers. The important thing is to win the popularity contest, having a better model of reality is only incidental. The thing to protect is the pleasure of winning, but other people's applause lights can be used strategically.

A person from X1 has only friends in X1 and X2. A person from X2 has friends in X1, X2, Y2. Assuming that having more friends is an advantage, the mutual politeness creates an advantage for people from X2 and Y2, and this is why they are doing it. I'd call that cooperation. In their case, cooperation is both a strategy and a goal.

In a way, also people from X1 and Y1 cooperate, but this cooperation is purely instrumental, as they hate each other. However, any act that successfully increases the mutual hate between groups X and Y helps them both, because it reduces their relative disadvantage against the 2.