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There is no glitch that just adds trash labels. Did you check if you have any filters set up that might be responsible?

I did. I cleared out a few filters even though I am 95% sure they wouldn't match the emails. Wouldn't be the first time I was wrong though.

The main thing I find irritating is that it contains the Inbox label as well.

Any filter to send something to the trash says "Skip Inbox, Delete It".