  We consider (self-adjoint) families of infinite matrices of noncommutative
random variables such that the joint distribution of their entries is invariant
under conjugation by a free quantum group. For the free orthogonal and
hyperoctahedral groups, we obtain complete characterizations of the invariant
families in terms of an operator-valued $R$-cyclicity condition. This is a
surprising contrast with the Aldous-Hoover characterization of jointly
exchangeable arrays.
