vice versa
Two words, no hyphen, and no stray a in the middle. It’s Latin for the other way round.
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Wrong: She trusts him, and vice-a versa.
Right: She trusts him, and vice versa.
The extra syllable is something you hear, not something anybody writes.
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Wrong: I cover mornings, he covers nights, and visa versa.
Right: I cover mornings, he covers nights, and vice versa.
A visa is a stamp in a passport. It is not going to help you here.
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Wrong: The rule applies both ways, and vise versa.
Right: The rule applies both ways, and vice versa.
A vise is a clamp bolted to a workbench.
Where it comes from
Latin, and it is doing more grammatical work than it looks like. Vice is the ablative of vicis, meaning a turn or a change—the same vice that gives you a vice president, the person standing in the place of someone else. Versa is the feminine past participle of vertere, to turn. Put together: the position having been turned.
So vice versa means the positions are swapped. English borrowed the whole phrase intact around 1600 and never bothered to translate it, which is why it still looks foreign and gets handled like something breakable.
Why the wrong one survives
Because of how it sounds. Most people say it with three syllables—VY-suh VER-suh—and that middle schwa lands exactly where the English article a would go. Write down what you hear and you get vice-a versa, which looks almost plausible if you have never seen the phrase in print.
The other two are eye-slips. Visa and vise are both common words that sit one letter away, so autocorrect leaves them alone and your eye skates right over them.
And versa vice is just the phrase getting turned around, which is at least thematically consistent.
In fairness
Both pronunciations are correct. VY-suh VER-suh and the flatter two-syllable vice versa are equally standard, and dictionaries list both without comment. Say it however you like—just don’t spell the extra syllable.
The hyphenated vice-versa turns up in older British printing and is not exactly an error, but it is out of fashion everywhere now. Two words, no hyphen.
Two words. Neither of them is a.