Well... different ranking outcomes (different sides of the coin) are possible. Just that the interpretation will always be "don't reject the null hypothesis" but yeah. :)

Either way, my overall reaction to your post is "yuck" (not your post itself! That I upvoted. I mean the whole situation... That a relatively standard statistical test could allow this sort of madness. I mean, I know frequentist stats isn't the Bayesian way, but that relatively standard methods in it can be this pathological does not at all give me warm fuzzies)