This sounds like you may have an inbox rule set up on the server that is going haywire. Or, it's possible you have an inbox rule set up on the server and the server itself is going haywire. Those are the first two things I'd look at. I'm guessing by your post count that you understand the nuanced difference between an inbox rule on the server vs an inbox rule in your desktop mail client - that said, it's something easy to overlook as well.
I recently had an issue once where I set up an inbox rule on the server to text me (forward to my phone number @ texting email address) whenever a keyword went to my email. The server had an issue where it would reboot overnight, resending all my texts. It sounds super similar to what you're describing and is something easily overlooked.
Are you able to confirm anything about inbox rules and/or if you have ninja-setup rules on the server by accident?