What I understood here is that Jager build the movement, and bought the system of the case from Mr Alfred Chauvot, in 1931....
Here is the Reverso Patent:
You're absolutely right, in the 30's, some other brands made some Reversos watches, like Patek Philip, and Movado.
Movado
Don't know who to credit...)
Look at the pics taken by Dan 16 in the PP forum, you'll see a PP Reverso!
Best.
Nicolas