OK, so I extended my little program to handle non-triangular ridges as well, and so here's a little animation of a projection of the tetracube into 3D:
This is, of course, an unconventional viewpoint, but this is more typical of what a 4D being would see. (The traditional cube-within-a-cube view is, of course, the analog of staring into a cube face on and seeing a square. Not exactly how most people imagine a cube.) Of course, only 4 of the 8 facets are actually visible, because the other 4 are obscured. This makes the individual facets a lot more easily discerned.
Now, if I can only twist my head around generating the face lattice of the 120-cell, ...