A quick sketch in Google Sketchup gives me this:
The central slice would turn in 30 degree increments. All the equatorial vertices have equal angles. It really looks like an expanded version of the Mixup Kites.
Aren't Witeden working on a couple of Mixups
where the central slice is 12-fold (turns in 30 degree increments?) I remember making mockups of 30-degree mixups with various cut depths, trying to make a mechanism of sorts in SketchUp and seeing Witeden's working on the exact same puzzle the day after. If I remember correctly this was one of the cut depths they had.
I'm not really sure if the polyhedron I made would retain its shape with those cuts, though.