When we draw all the diagonals of all the faces of a cube, we obtain the edges of two regular tetrahedra. Keeping one of these fixed, we move the other under the condition that the originally intersecting edges of the two tetrahedra should still remain coplanar (i.e. intersect, are parallel or coincide). We determine all such finite motions. These will be seen to constitute one-dimensional and two-dimensional smooth manifolds. We also deal with the infinitesimal degrees of freedom of the motions of our mechanism. In several positions the number of infinitesimal degrees of freedom is not one or two, but is three; this is connected with the bifurcation phenomenon of the solutions.