  We characterize polynomials having the same set of nonzero cyclic resultants.
Generically, for a polynomial $f$ of degree $d$, there are exactly $2^{d-1}$
distinct degree $d$ polynomials with the same set of cyclic resultants as $f$.
However, in the generic monic case, degree $d$ polynomials are uniquely
determined by their cyclic resultants. Moreover, two reciprocal
(``palindromic'') polynomials giving rise to the same set of nonzero cyclic
resultants are equal. In the process, we also prove a unique factorization
result in semigroup algebras involving products of binomials. Finally, we
discuss how our results yield algorithms for explicit reconstruction of
polynomials from their cyclic resultants.
