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My preferred visualization of the Complex 3x3 requires a turn that looks like {1, 2, 4}. This requires the axis to affect a disjoint region. Allowing nonconvex polytopes will also allow the puzzle to be super-stickered without requiring piece-explode.
The 3D/"physical" version of the Hemimegaminx, likewise, requires a turn of {1, 3}.
Examples:
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