Turning in four dimensions

Drag θ and the tesseract turns in the xw-plane, and its shadow turns inside out: the small cube in the middle swells outwards while the outer one shrinks in. The coral square lies in the yz-plane, and however far θ goes, it never moves within the shadow: when a turn happens in just one plane, the plane at right angles to it stays still. At about 200° the gold cell's shadow is squashed almost flat and its volume in the corner falls close to 0; then it opens out again, inside out, and swells. In the shadow the edges grow and shrink, but their length measured in 4D never changes, and the vertices, edges, faces and cells counted in 4D are all still there. Drag only φ and the shadow turns as a whole, like an ordinary solid, without changing shape; with θ and φ both away from 0 (and from a whole turn), the turns in two perpendicular planes combine and every point but the centre leaves its place. Every turn in four dimensions can be written like this, for some pair of perpendicular planes, each turning through its own angle. Try the other polytopes too.