Is it possible for the black wire to be hot but to create reverse polarity in a GFCI?
As I've said elsewhere on this site, I find no-contact voltage testers to be unreliable. I've seen both false positives and false negatives with mine, especially in the presence of multiple conductors, some of which are hot. The only thing I would use a no-contact voltage tester for is to confirm a condition I already know, and even then I would only use it to test if an entire circuit is live or not, not individual conductors.

If you are getting confusing readings, the first thing I would do is use a proper multimeter to get reliable voltages, then you can go from there.

But to answer your question: no, the GFCI should not be reversing the polarity of the circuit. Are you sure you're wiring up the GFCI correctly? Most (all?) GFCI outlets have different "LINE" and "LOAD" terminals, which are you using?