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I commented wondering whether it's possible to build what amounts to a Rubik's UFO which only supports 180 degree turns on the horizontal axis. It turns out that it is, using a novel and interesting mechanism, vaguely analogous to the half turn only cube.
The core consists of two pieces connected on a horizontal axis which they can spin freely on. Around the equator there are six circles of three different sizes with opposite ones being the same size. They're evenly spaced and small enough to not overlap. The equator line divides the circles in half, so there's a total of twelve pieces added. One half of the core has every other semicircle glued to it, to make the eventual deep cutting work. The next shell up holds down all the semicircles and equalizes their sizes, and shelling proceeds in the standard way from there until the outermost shell is deep cut.
Manipulating this puzzle might be a bit strange because there's no visual indicator of when the non-horizontal axes are unblocked, but it is in fact a doctrinaire puzzle, and might be an interesting solving experience, and is certainly a novel mechanism.