I'm playing a FighterMage that uses mostly fighter skills, but some magic skills for support (alteration/mysticism/illusion/alchemy). I joined the Mages guild and the Fighters guild. I know the mages guild and House Telvanni hate each other, so it would make sense to join both of those. Does it make sense to join both Fighters guild and House Redoran? Do they hate each other in any way? Are they compatible?

Thanks

I played many times as a fighter and each time I joined Mages and Fighter's guilds and:

Once it was with House Redoran. Then House Hlaalu and in another play House Telvanni. Always together with the 2 previous guilds.

I never found any problems.

If you're doing a strict RP, maybe avoiding some combinations makes sense but if it's just a matter of practical gameplay, the disposition penalty makes almost no difference - just a few points that a raise in rank will make up for.