So many answers in here about "fluid dynamics"

The developing and fixing reactions are not anywhere near complicated enough to be significantly affected by any sort of agitation provided you aren't shaking your developer like a martini. Even a gentle swirl with the twizzle stick is going to expose all parts of the film to fresh developer.
I use a twizzle stick when I'm doing c41, I leave the tank in my water bath so the temperature doesn't drop, you can't really do inversions underwater. When I do Rodinal I do inversions sometimes if I'm using a low concentration and the dev times are long. 90% of the time I stick to twizzle sticks because they're cleaner. 30+ y/o tanks tend to drip a bit.