Oh, wow. That's interesting.
I think the lightbulb is starting to flicker on (finally). So, for the stance/hanmi, gyaku means "the other way" (as opposed to the normal way) rather than the positional/geometric opposite, and for the grab gyaku means "the other side" (cross hand)? That could make sense.
Am I getting closer to having a clue?
Since your explanation is more confusing to me than how we use the terms in my own dojo, I think I'll bow out of this discussion. I think the best course would be for someone from your own dojo, who understands your dojo's use of the terms, to show you directly.