This setup has been in my head for awhile, but I haven't been able to try it out!

We made an attempt at multi-sided flats, seemed like a great idea at the time, but found that the rake of the stage is too deep. When you turned the piece it tends to want to tip over.

As suggested by Tim a revolving Periaktoi flat unit, usually a 3 flat system also used in multiple Periaktos scene systems. However double & quad sided units were used by the ancient Greeks & even in Chinese theatre.
As mentioned a flat stage deck makes it much easier to work the system.
I have made a few Peri or Medicci flats, as legs or a multi backing system on a pulley drive. Much the same as the revolving advertising periaktos used around the place.
The easiest is the single double side flat with a pivoting central spigot on the deck.
The TV flats being rigid are better, although it works just as well using the soft skin flats.
The split loose pin hinges make it easier for storage & the use of single flats.
If the removable pin hinges are not readily available off the shelf, it is easy to make up your own.
Grab almost any butt hinge & grind off the pins captivating head, separating the to halves. I find the square butt hinges are better as the rolled pin retainers are equal on both halves, however any type ill do.
Get some steel rod the same diameter as the pin & add about 2" to the length of the pin size,  to roll bend one end as a finger pull to assist in removing & inserting the  pin.
If just a deck spigot is used to revolve
with the flats on the floor. Using old carpet strips tacked, glued or stapled upside down onto the bottom rail of the flat works as  well as castors. Therefore there is no need for a skirt to hide the castors.