Is there a word to describe an exactly opposite situation? In context, if someone describes a situation in which he found himself, and I tell him I found myself in the exactly opposite situation, what word would describe this? Let's assume irony may be involved in some way, but that is not the word I am looking for.
Since you have used the word irony, I am tempted to say that two exactly opposite situations are not ironic; they are coincidental. Perhaps an example would help.
Frieda: "My dog Willie ran away this week, and I haven't been able to locate him."
Sally to Freida: "Hmmm. I suppose I shouldn't tell you, but just this week I became the owner of a rescue dog from the Humane Society!"
Here are two "exactly opposite situations," but they do not necessarily involve irony, just an unfortunate coincidence. One situation is sad; the other is joyful. One person lost a dog, and the other person "found" a dog.
Perhaps commenters @WS2 and @Kris, above, are on the right track in suggesting that what you describe as "exactly opposite situations" are in some sense "antithetical situations."
Is there a single word besides antithetical, though? That's a tough one. Two words would work just fine, but one word? Not so much! How about
a weird coincidence?
an incredible coincidence?
a surprising coincidence?
a sad coincidence?
a snafu? (s ituation n ormal, a ll f ouled u p)
bad karma/good karma?
bad luck? (A thief steals a hundred times in a row without getting caught, and you, when you impulsively and for the first time in your life pick up a donut in a store and eat it without paying for it, get caught and arrested for shoplifting! Now those two situations are in a sense "opposite" and tinged with irony, especially if you have a reputation for being an upstanding, law-abiding citizen.)
Unless you provide a clearer context for your two "opposite situations," I'm at a loss as to what to call them, at least in one word.