Although I haven't tried it, this presumably would be easier to see and do with a small and large tetrahedron rather than a cube. Eugene Salamin writes:
I've seen this phenomenon described in a nice way, although I haven't done the construction myself. Build a frame, say a cube with 12 rods. Place an object inside the frame, and tie it with strings to the frame, for example 8 strings to the vertices of the cube. The strings need to be slack enough to provide the wiggle room required. Rotate the object 720 degrees. It is then possible to manipulate the strings while holding the frame and object fixed, so as to untwist the strings and restore the original configuration. On the other hand, if the object is rotated 360 degrees, this manipulation cannot be performed.