Depends. Is TerraFist ok with getting blamed and taking credit for acts not theirs? Some real world groups take credit for anything they can. And will TerraFist be ok with a shooting match with mtc and likely losing those operatives?
I think you last paragraph gives enough justification for them to show up just because you think it'll make your game better. And in most cases your players won't notice a slight inconsistency in ethos of that group.
It also kind of reminds me of the wacky plots of something like metal gear solid: "The eco terrorists aren't the ones who took over the oil trig after all? But then they showed up anyways? What the hell is going on?" Good times for all.