Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function $ζ(s)$. Their approach indicates how lower twisted moments of $ζ(s)$ may be used to evaluate higher moments. In this paper, we present a rigorous random matrix theory analogue of their heuristic. To do this, we develop a notion of "twisted moment" of characteristic polynomials of matrices in the unitary group $U(N)$, and we prove several identities involving Schur polynomials. Our results may be viewed as a proof of concept of the heuristic for $ζ(s)$.