There's realistically only 3 (maybe 4) possible outcomes:

They reach the the "latitude of no return" (I think it's 40th?) and:

1) Speed up and completely flip,

2) "bounce" off the latitude and start going back, staying in their respective regions,

3) Stop at the latitude and slowly rotate around the latitude axis

Since we have NO clue what caused Earth's angular setting, it's entirely possible these changes are not normal and are residual effects of something that happened long ago (ie: a comet impact, solar flare destabilizing the core, etc). For instance - the Chinese myth of Nuwa and Gong Gong tells of a battle between gods, where one was thrown into a mountain, which caused the earth to tilt, created a huge flood and manifested the seasons. How is it ancient myth lines up with established understanding?

Arthur Koestler once said "myths, like crystals, grow according to their own recursive design - but both require a real basis".

The "mythical" fourth outcome would be something akin to them "dying"... not a good thing, but that would involve the solidification of the core and that just seems a bit... too far fetched, but it's still a logical possibility. (Some have even speculated the sun is far older than we are told - and that it itself could also be, dying.)

Out of the options, only #2 doesn't lead to a whole hell of a lot of problems (mainly regarding EM technology, GPS, probably animals (as they use magnetic fields to navigate, etc) (likely humans too)).

I've been watching this situation for a while now - there's a channel called "Mavstar Observatory" who has been assembling field monitors and shipping them around the world, to get a picture of what's happening.