> Are there no twin primes (p,p') where p%3 == 1 and p'%3 == 2?

By definition, no there aren't. Twin primes are a distance of 2 apart, so your mod 3 options are 0 -> 2, 1 -> 0, and 2 -> 1. Anything involving 0 means it's divisible by 3, so your only mod 3 possibility for twin primes is 2 -> 1.

EDIT: Excluding the possibility where 3 mod 3 = 0, of course. This allows a 0 -> 2 transition with (3,5).