Alright, as I promised, here is a projection of the 600-cell into 3-space, with hidden cells removed:
To show the intricacy of the inner structure more, I've highlighted the 20 tetrahedra surrounding the central vertex (which is actually the closest vertex to the 4D viewpoint). As you can see, they are joined in an icosahedral configuration. Two other layers of tetrahedra are visible, wrapped around this central icosahedron. It's a bit hard to discern individual cells because of the limitations of a 2D screen. Maybe I should make some static images with different cells highlighted.
And now, a dream come true: the 120-cell!
The closest cell to the 4D viewpoint has been highlighted, to make it easier to discern the structure of this projection. As you can see, the central dodecahedron is surrounded by a layer of 12 dodecahedra. You can barely see some cells from a 3rd layer (the outer hexagonal faces are actually dodecahedra seen from a 90° angle, so they appear flat). The rest of the cells are obscured from the 4D viewpoint. Quite interestingly, this particular 4D viewpoint gives us a projection envelope in the shape of a buckyball (truncated icosahedron).
Note that these projections are rotating only in 3D; the 4D viewpoint is fixed. I'm going to work on rotating the 4D viewpoint as well.