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Indeed, I thought at the early part of this episode that the only thing stopping this episode from getting a ten was the lack of Amy Acker... and she hadn't even been born in 1967, so they had a good reason, allowing me to give it maximum marks... That wasn't quite the case at the end of this.
After diplomatic negotiations to obtain dilithium crystals from a peace-loving planet go badly, Kirk and his landing party, made up of McCoy, Uhura and Scotty, beam back to the Enterprise... during an ion storm.
This ion storm results in them not going to their intended destination, but to a rather different universe in which the Federation is instead an Empire with a generally ruthless approach to obtaining resources. As well as a command structure in which promotion is attained by assassination...
Kirk and his team must make their way back to their own universe before time runs out or someone kills them. Also, their evil counterparts have ended up in their own universe...
It's fair to say that has been a highly influential episode of the show; the 'Mirror Universe' appeared in no less than five episodes of Deep Space Nine and one of Enterprise. Mirror Universe uniforms are a purchasable costume choice in Star Trek Online, also featuring in three missions there. AJJE used to have a Mirror Universe-set sim itself as well...
This Mirror Universe - inspired by episode writer Jerome Bixby's own 1954 short story - has also been the source of much homage (Doctor Who did it in "Inferno" way back in 1970, Fringe made a major arc of it and Buffy naturally had a go as well) and much parody; when you're the subject of a skit on Mystery Science Theater 3000, you know you've made it. The list of these references is very long...
One particular element that is much parodied is Mirror Spock's goatee - there is even a band called Spock's Beard. Having a beard be a sign that you're a bad guy (rarely a bad girl) is a long standing trope of the genre going back into the 16th century; Satan has long been depicted with facial hair. That said, this version of Spock actually proves to be one of the nicer characters on the ISS Enterprise. Less forgiveable, however, is having the sexual deviant Mirror Sulu have a massive scar down his face.
Another common plot device is that when characters turn evil, they wear less clothing, especially if they're a red-headed female Canadian with a body that can pull off a cropped green camisole/print shorts combo. The landing party not only swap places with their counterparts... they also end up in their clothing. Scotty and McCoy, being not exactly sex symbols by conventional definition, just wear their regular uniforms with different badges. Kirk, as leading man, ends up in a sleveless top showing off his muscles... and Uhura wears a midriff baring outfit. To cap the latter off, she later does the whole 'sexy distraction' shtick on Mirror Sulu.
All that said, the Mirror Universe is a very interesting place - although I wouldn't want to live there - with plot elements that would inspire many works. Particularly notable is the 'agonizer', used to punish wayward crew with pain. In its hand-held form, it rather prefigures Root's favourite weapon, the Taser, which hadn't even started development at this point!
(Have I mentioned that I'd really like Acker to make an appearance in Star Trek: Discovery?)
This episode is a tour de force for all the regulars; many of whom get to play against their regular types. 'Our' Kirk is brilliant - realising very quickly that he is going to act different enough to avoid suspicion while still trying to save a planet from destruction by his counterparts. However, McCoy is also great, deciding to save Mirror Spock from death in the climax despite the fact it would put him in peril.
Mirror Kirk also has a mistress, a female officer we've never seen before... and who Prime Kirk (sounds like a brand of beef that) doesn't recognise, although he does snog her. It turns out in the (actually fairly good) final scene that she's a freshly assigned officer arrived a week before... shouldn't the Captain meet his new officers on day one?
Star Trek can be accused of cliché in many cases when in fact it wrote them. Unfortunately in this case, there are some pretty unoriginal plot devices in this episode.
However, that does not detract from what is a very good tale and one that continues to have a major influence.
Just a pity it couldn't have a gun toting Amy Acker in it.