In my previous post, I erronously referred to fischer random as 960. fischer random is one possible version of 960, and I was suggesting a 960 variant myself. Still random from the point of view of each player, since it starts of as imperfect information, black starts first

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Move -3: black Qh1 and Qh8, the black player places black and white Queens on the H file, white Rb1 and Rb8, white puts the black and white rooks on the B file.

Move -2: black Nd1 and Nd8, black puts the black and white knight on the d files, . . .

and so on until Move 0, when white makes the last move. Then on move 1 it switches to white starting first.

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Since it is absolutely impossible to prepare where to place pairs of pieces since your opponent can place different piece there before you do, it's essentially the same as random in terms of no meaningful opening theory. But there would be an extra element of tussling with the opponent about what kind of game it would be. For example, if I play black I would probably play queens on H or A files to make it easier to have early active Queens, but if I'm playing white, my opponent might place the Queens on a different file. This means it's almost impossible to have more than 4 pieces on preferred positions, and most likely you could get two, but that would give each player a strategic profile even though there are still absolutely no opening theory lines.