Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph and asked under what conditions such a polynomial is even-interpolating, odd-interpolating, or both. In this paper, we provide an answer to this open problem.Using the framework of delta-matroids, we prove that the twist polynomial of any binary delta-matroid is either an even polynomial, an odd polynomial, or both even-interpolating and odd-interpolating. Applying this to ribbon graphs, we deduce that the partial-dual polynomial of any ribbon graph satisfies the same conclusion.