  We consider the existence of a continuous set of mutually unbiased bases for
the continuous and periodic degree of freedom that describes motion on a circle
(rotor degree of freedom). By a singular mapping of the circle to the line, we
find a first, but somewhat unsatisfactory, continuous set which does not relate
to an underlying Heisenberg pair of complementary observables. Then, by a
nonsingular mapping of the discrete angular momentum basis of the rotor onto
the Fock basis for linear motion, we construct such a Heisenberg pair for the
rotor and use it to obtain a second, fully satisfactory, set of mutually
unbiased bases.
