I think I got it:
gem5.jpg [ 131.6 KiB | Viewed 837 times ]
This is essentially a Master Tetrahedron (Master Skewb in Tetra shape).
The black lines show the cuts. The green lines are Truncations. After that the blue lines indicate the second truncation. That of the octahedron's corners. The shallower cuts are done in a way that the square faces formed by that second truncation split in the middle.
Building up on those surfaces would result in new parts. To be exact, one additional part between the intersecting cut planes on each square face. Those parts are the truncated octahedrons tips.
I hope I am right with all this.
EDIT: NO, this is wrong. Something is wrong with the blue lines. A truncation there wouldn result in square faces that are split in the middle. The cutplanes would go through the resulting edges of those faces.