  The classical problem of counting the number of real zeros of a real
polynomial was solved a long time ago by Sturm. The analogous problem of
counting the number of zeros that a polynomial has on the unit circle is,
however, still an open problem. In this paper, we show that the second problem
can be reduced to the first one through the use of a suitable pair of M\"obius
transformations - often called Cayley transformations - that have the property
of mapping the unit circle onto the real line and vice versa. Although the
method applies to arbitrary complex polynomials, we discuss in detail several
classes of polynomials with symmetric zeros as, for instance, the cases of
self-conjugate, self-adjoint, self-inversive, self-reciprocal and
skew-reciprocal polynomials. Finally, an application of this method to Salem
polynomials and to polynomials with small Mahler measure is also discussed.
