An oxymoron is a word or phrase that is contradictory. My favorite is jumbo shrimp–although I’m not sure that this is actually contradictory. You have your small shrimp. You have your medium shrimp. You have your large shrimp. Finally, you have your jumbo shrimp. I guess it’s contradictory because how jumbo could a shrimp be? Anyway, I’m not here to talk about shrimp.
In Literally, the Best Language Book Ever, I wrote about the redundancy of exact same or exactly the same. (I don’t know why I didn’t write about jumbo shrimp–maybe volume 2.) Exact opposite is the counter to exact same, except rather than a redundancy, it’s an oxymoron.
Exact means the same and opposite means different, so they shouldn’t be paired–unless you like sounding like a moron–well, at least an oxymoron.
–Paul
Paul’s book–Literally, the Best Language Book Ever
Everything Language and Grammar 
How about a hyphen between the words? “Exact-opposite?”
What would be a better phrase? You assume that exact opposite means “a mirror image” – as in “exactly inverse” or “equally opposite” – Is there a better set of two words that would convey that meaning?
Reply from Paul: The hyphen doesn’t help, and I think that “opposite” is what people mean, so that’s what they should say. They add “exact” to it, thinking that it will add dramatic effect, but it just adds another word—and one that is both superfluous and contradicts the meaning to some degree.
Exactly the same may be an oxymoron, but it’s use of often not for clarity, more for impact, especially when spoken. He walked exactly the same way and He walked the same way may technically say the same thing, but the former stresses the similarity more clearly.
Reply from Paul: Yes, the redundancy could be a deliberate choice to add effect.