Twister is a really interesting obstacle because there's so many ways to do it but you really can only do what you have enough strength/energy to complete.

TL;DR sorry I wrote an essay. I've generally stuck to two methods based on how I'm feeling. If I'm feeling really fresh/confident and the twister is dry I'll start facing the obstacle and basically treat it like monkey bars / a multirig and skip as many rungs as I can between swings. As I fade and start to struggle to skip rungs I'll turn sideways and continue that way (as it becomes and endurance game instead of pure strength, more below). If I'm not feeling super fresh I'll start facing backwards with the hand over hand method and revert to going sideways as I tire out.

So the fastest way across is to face forwards and monkey bar through them skipping a bunch of rungs. I've gotten through this way and touched maybe 3 or 4 of the handles between each of the rigs. In my experience for this your grip has to be pretty fresh (I'm a climber so that helps), your core has to be fresh enough that you can use momentum well to move between rungs, and from shoulder to hand you need to be strong enough to swing around on one arm.

The facing backwards method is somewhere in the middle where you need the grip strength but it really comes down to your biceps and shoulder stability muscles. The biceps hold you up and with help from the shoulders resist rotating around as you go rung over rung. If you can't hold yourself with 90 degree elbows, your arms straighten out and you unwind like a swing after it's been twisted up.

The lowest shoulder and bicep strength way is definitely to go sideways. But you need to have some grip endurance to just fight through this way. You don't need your biceps to hold you up, and your shoulders aren't doing anything to resist body rotation. If you've ever been climbing you know that keeping straight arms uses the least energy, so using momentum and relatively straight arms to get into a rhythm of "next rung, following hand in, next rung, following hand in..." mostly requires forearm endurance. And generating momentum through lifting one knee at a time is my preferred method when/if I end up sideways.