Forgot to mention how they get them.

Either someone has my login and is messing with me by somehow adding the Trash and Inbox label or there is a glitch in the system putting both labels on it. My guess is the glitch.

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".

I go into the trash label inside Gmail. I click on a message that I didn't want to go straight to my trash. At the top where it shows the Trash label there is also an Inbox one with it.

As soon as I click the "x" on the label, I can see the email being downloaded by the mail client I use. It is of course displaying the Inbox and not Trash.