We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial of degree , there are exactly distinct degree polynomials with the same set of cyclic resultants as . However, in the generic monic case, degree 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.