Exercise: Note that both the shape of the compound of five tetrahedra and the color pattern of the five-colored icosahedron are chiral. If we reflect a five-colored compound of five tetrahedra in a mirror, we are actually reflecting two things, the shape and the coloring. What results if we reflect just one of these two aspects? In other words, imagine assigning colors to the planes of the compound's shape according to the colors of the reflected colored icosahedron.
Answer: The result has no axes or planes of symmetry that preserve both the shape and the coloring. I call it an anti-colored model. (Perhaps there is an already existing word for it, but I don't recall ever seeing this idea discussed anywhere.) Notice that each tetrahedron, rather than being just in one color, is in four colors, and so omits just one color. All the planes of a common color define the planes of an imagined tetrahedron, but the tetrahedron so defined is not one of the tetrahedra in the physical structure.