I know "vice versa" more or less means "conversely," but when it is used by itself, should it be punctuated as if it were an independent clause?
Dogs don't like cats, and vice versa.
Dogs don't like cats and vice versa.
The Chicago Manual of Style is generally my guide in these things, but I don't have my paper version ready to hand, and the online edition is behind a paywall.
However, they do have a handy Q&A section which is freely available. Do they have a definitive question and answer for you? No, they do not... but they do have quite a number of answers that contain the phrase. Out of six hits on a search for "vice versa", five of them are separated by a comma, and only one is not.
With a comma:
Without a comma:
Use a comma before "and/or vice versa".
From Why Learn to Punctuate?:
Since the above was part of an article talking about how to punctuate, I believe it will carry more weight.