Back in the 1860s "reversible" frames were the thing. Some people just flipped the boxes upside down. Some had frames where there was a pivoting frame inside of the frame so you could flip the comb upside down. The queen laid in it fine. The point was to force the bees to rearrange the brood nest constantly to prevent swarming. The queen seemed to have no qualms about laying in them. Then there is the guy with the hoop shaped frames that rotate in the hive for Varroa control. The queen seems to have no problem laying in them either. I don't think it really matters all that much.
In the 1880s, when the Heddon hive had been recently patented and there was much discussion about whether multiple shallow brood boxes had already been done (Heddon's hive was an eight frame hive with combs about 4 1/2" deep or so making it similar to two eight frame shallows), whether thumbscrews to anchor the frames so the box could be inverted had been done by others and whether inverting the frames had been done. None seem to contest that Heddon had invented a frame that allowed the comb to be inverted (a frame inside a frame that swiveled).
They would purposely invert the combs in the brood nest to get the bees to move the honey cap out, prevent swarming and expand the brood nest. Many people have said that queens cannot lay in inverted combs, but apparently many people were doing this and there was no mention of any reluctance on the part of the queen to lay in the brood nest, in fact it was done to get her to expand the brood nest. There was also no mention of any problems storing or moving honey with inverted .
Back to the man with the round combs that rotate, in order to control Varroa. I'm not that interested in doing that much work myself, but if someone wanted to experiment with the idea, inverting brood boxes should have a similar effect on the Varroa. If one inverted the brood boxes every six or seven days then all the Varroa would have to deal with the inversion during the course of their stay in the sealed brood.
Here are a couple of quotes from books of the time:
"While the reversing of brood combs will produce no ill effects whatever, numerous are the advantages arising from such reversal; some of which aid us materially in accomplishing the desired results which are partially accomplished in the contracting system, above described.
"When using frames even no deeper than the standard Langstroth, you know how the bees (especially Italians) will persist in crowding the queen by storing honey that ought to go into the surplus department, along the upper edge of the brood combs, just under the top bar, and farther down in the upper corners, till by actual measurement we find that nearly one-fourth of each frame, and sometimes more, is occupied with honey.
"Now if we reverse the frame containing a comb so tilled, we place the honey in an unusual position; in a place usually occupied with brood, and when this is done in the breeding season, when the bees are not inclined to decrease their quantity of brood, this honey will be immediately removed to the surplus department, and soon the frame will be one solid sheet of brood, which is a glad sight to the bee-keeper whose experience has taught him the value of a compact brood nest, free from honey."
Success in Beeculture by James Heddon Pg 85
It seemed a pretty common subject:
"REVERSIBLE BROOD FRAMES.
"The engraving represents the reversible brood-frame made by Mr. James Heddon. Many devices have been presented to reverse the frames, but this is as good as any, where reversing is desired."
Bees and honey, or, The management of an apiary for pleasure and profit by Thomas G. Newman pg 44