The implications of constraints on unitary transformations of spin operators with respect to the accessible regions of Liouville space are analyzed. Specifically, the effects of spin-permutation symmetry on the unitary propagators are investigated. The influence of 2 and 3 propagator symmetry on two-dimensional bounds for F z = Σ N i =1 I iz ↔ G z = Σ M j =1 S jz polarization transfer in IS and I 2 S spin- systems is examined in detail. One result is that the maximum achievable F z ↔ G z polarization transfer is not reduced by permutation symmetry among the spins. For I 2 S spin systems, 3 symmetry in the unitary propagator is shown to significantly reduce the accessible region in the 2D F z -S z Liouville subspace compared to the case restricted by unitarity alone. That result is compared with transformations under symmetric dipolar and scalar J coupling as well as shift and RF interactions. An important practical implication is that the refined spin thermodynamic theory of Levitt, Suter, and Ernst ( J. Chem. Phys. 84 , 4243, 1986) for cross polarization in solid-state NMR does not predict experimental outcomes incompatible with constraints of unitarity and spin-permutation symmetry.