We lift the well-known but mysterious random-to-random shuffling operators of Reiner–Saliola–Welker from Markov chains on symmetric groups to Markov chains on Type-A Iwahori–Hecke algebras. It turns out that this family of operators still pairwise commute, and that their eigenvalues are polynomials in q with non-negative integer coefficients. Our proofs use tools from the representation theory of Hecke algebras and center on a key recursion relation involving the Young–Jucys–Murphy elements. This q-perspective both generalizes and simplifies the results of Reiner–Saliola–Welker and Lafrenière for the symmetric group. (This is joint work with Sarah Brauner, Patricia Commins, and Darij Grinberg.)