  We show that if a real trigonometric polynomial has few real roots, then the
trigonometric polynomial obtained by writing the coefficients in reverse order
must have many real roots. This is used to show that a class of random
trigonometric polynomials has, on average, many real roots. In the case that
the coefficients of a real trigonometric polynomial are independently and
identically distributed, but with no other assumptions on the distribution, the
expected fraction of real zeros is at least one-half. This result is best
possible.
