Pardon my ignorance.
If your back has a square shape, why don't make it with a slot above, and a slot on the side?
You don't need rotating the back.
You would insert the sheet film from above for a portrait shot and from the side for a landscape shot.
Fixed "guides" would centre the sheet on the lens.
Considering that the back is square, and the sheet film is rectangular, the guides for portrait orientation would not interfere with landscape orientation and vice versa.
Am I missing anything? (never used a LF camera).