Just recently picked up an old Peavey Deuce as it's the closest thing I have found to a Twin in my price range. Had the same issue described here. I pulled the reverb tank from my crappy 80s Fender Princeton Chorus and plugged it in, problem solved. Then I pulled the tank out of the deuce. It is clearly a new tank, so I was surprised to find that inside it there were blocks of light foam (think packaging for a retail product) in the center of the tank. Pulled those out and the issue was immediately resolved.
My theory here is that the blocks were acting the same way your finger does when you play the harmonic on a guitar string at the fifth, seventh, 12th (and so on) frets of a guitar.
Wish my Deuce were your Twin, but now that it doesn't feedback it is exactly what I was looking for.