One of my short list--that handful of episodes I love so much I prefer not to rank them. It is the perfect example of Trek's gift for straight-faced camp: sure, the idea makes no sense, really (as The Laughing Vulcan once astutely demonstrated) and the things that make it great (see scotpens's illustrated list above) all are silly as hell when looked at from a certain (killjoy) angle but the show sells it so that it feels like serious SF. Love it.
There are a bunch of great episodes of TOS whose premises, logically and scientifically, make no sense at all.

But does it really matter? I don't think of
Mirror, Mirror as just a silly campy romp where people dress differently and get to wear beards or scars (and I think that DS9 Mirror Universe episodes went wrong the moment that the writers seemed to believe that this is all that the MU is about), I like it because it is a look at a dark version of the Federation. It's an interesting concept to explore - that in a different set of circumstances, our heroes might have turned completely different, and that a united Earth could have become a leader of a barbaric, ruthless conquering empire.