I know this is a physics paper and not a math one but as a mathematical physicist / relativist I find it very confusing that you treat "orientability" and "time-orientability" as synonyms. These are completely different concepts, though, see e.g.
I'll give it a shot. When you entangle two particles, it's kind of like they're connected to the same random number generator. So if you measure one particle it'll collapse to a certain value (at random), and you know that the other one will also collapse to the same value (or some other such correlation, depending on how you set up the entanglement). You also definitely absolutely cannot use entanglement to communicate information.
The way you described it, it seems like it could relay information:
If you see an odd value, meet me at Disney world.
If you see an even value, meet me at the grand canyon.
This way, our persuer won't know where we will meet.
Well, it relays information in the same way that each of you using a random number generator pre-initialized with the same seed relays information. It doesn't involve faster-than-light communication, is the point. Entanglement differs from classical RNGs in that there is no shared seed; no local hidden variable, as it's called. The measurements are just... correlated. If you're interested in a deep dive on entanglement I wrote a post about it: https://ahelwer.ca/post/2018-12-07-chsh/
https://physics.stackexchange.com/questions/14168/orientabil...