I have been working on the watch below, with a novel variation on the Reverso principle.

The actual watch case is pivoted on its center on two jointed bars top and bottom, which in turn have pins which run in grooves in the outer case.

To turn the watch part over you pull up on it until there is enough clearance to pivot the watch over in the desired direction.

And here is the 15J movement. Also shown are the sets of hinged arms on which the watch section turns, and the ends of the four arms showing the studs which move in two grooves in the outer case.