Seriously, what the heck? Is this normal for an employer to do?
No. As far as slimy tactics by employers go, this is pretty up there. If I were you I'd name & shame them on glassdoor and the like. If they're in any sort of public spotlight the PR from that will be disastrous. Nobody wants to interview much less work for a company with that track record.
Do I have any recourse against this employer?
Thats a question for a lawyer specializing in employment law. Try to find one that offers free consultations.
The more I think about, the more I'm thinking I must have said something offensive to somebody in an interview. Is there any way to ask them what I said (so that I can avoid this mistake in the future)?
Even if you did, I have never heard of a employer doing this, and I'm pretty sure almost nobody actually does this. If you are an employer flying people out for interviews, you CAN NOT afford the bad publicity such a stunt will give you. Imagine what happens if a company like that invites me to an interview and I read THAT on a review site? Hard pass just on precaution.