Consider a mechanical system consisting of two completely symmetric, three-dimensional rigid bodies, each with inertia tensor the identity matrix and mass center at 0 is-an-element-of R3. Imposing the constraint that this system have zero total angular momentum, the angular velocities of these bodies are negatives of one another, and the transfer of this system from one position to another is nonholonomic. While Chow’s theorem establishes the fixed-endpoint controllability of this system, this result does not explicitly exhibit any motions between a given set of endpoints. This paper shows how to explicitly construct simple motions of this system on [0, 1] with arbitrary endpoints in SO(3)2. In particular, using normalized quaternions to describe rotations in terms of elementary functions, a continuous motion with the given endpoints is constructed; it consists of at most three successive motions during each of which the bodies rotate on fixed axes.