I would guess I could take a typical box with the frames reasonably tight and roll it (in line with the frames) until it's upside down...
If I understand the theory right, you won't need to turn it upside down. I think it may be sufficient to turn it 90 degrees (in fact, if you turn it 90 degrees to one side, and then 90 degrees to the other side, then you have 180 degrees). Of course, for a 90 degree rotation you'd have to invent a box that's almost as wide as it is high, and find a way to close off the "top" and "bottom" of the box when you've turned it 90 degrees, and find a way to allow bees to move up and down the hive even if the one box was turned 90 degrees.
The advantage of round comb is that you can rotate it without lifting the supers. If you want to "rotate" a box, you'd have to lift the supers every time you turn it.
Manually rotating the brood box won't help with varroa control (from a practical point of view), since varroa mites that successfully mate will lay one female egg every 25-30 hours, [
1][
2] so you'd have to perform the rotations more than once per day.
Just my opinion.