Recent work has established a uniform characterization of most 6D SCFTs in terms of generalized quivers with conformal matter. Compactification of the partial tensor branch deformation of these theories on a T2 leads to 4D N=2 SCFTs which are also generalized quivers. Taking products of bifundamental conformal matter operators, we present evidence that there are large R-charge sectors of the theory in which operator mixing is captured by a 1D spin chain Hamiltonian with operator scaling dimensions controlled by a perturbation series in inverse powers of the R-charge. We regulate the inherent divergences present in the 6D computations with the associated 5D Kaluza–Klein theory. In the case of 6D SCFTs obtained from M5-branes probing a C2/ZK singularity, we show that there is a class of operators where the leading order mixing effects are captured by the integrable Heisenberg XXXs=1/2 spin chain with open boundary conditions, and similar considerations hold for its T2 reduction to a 4D N=2 SCFT. In the case of M5-branes probing more general D- and E-type singularities where generalized quivers have conformal matter, we argue that similar mixing effects are captured by an integrable XXXs spin chain with s>1/2. We also briefly discuss some generalizations to other operator sectors as well as little string theories.