Relaxation of compressible plasmas to an equilibrium with flows is studied. The magnetohydrodynamic (MHD) equations with large parallel thermal conductivity and ergodic field lines show that ∫v⋅Bis an invariant even with compression. Also,S=∫&rgr; ln( p/&rgr;&ggr;) is the only entropy‐like invariant of the MHD equations with infinite parallel thermal conductivity;Sincreases in time with finite thermal conductivity. The other invariants are energy, helicity ∫A⋅B, mass ∫&rgr;, and possibly angular momentum. Equilibria are found by extremizing energy while conserving these invariants or by maximizing entropy with the energy and other invariants as constraints. These invariants are complete in the sense of generating all equilibria that form after relaxation with ergodic field lines. For parallel flows, there are three classes of solutions characterized by the sign ofd&rgr;/dB2and the mirror mode parameter. A sufficient condition for stability is derived. This condition is never satisfied by the class withd&rgr;/dB2>0, indicating the possibility of unstable resistive interchanges. The class withd&rgr;/dB2<0 is stable if generalizations of the local firehose and mirror criteria are satisfied, and if generalized Taylor helicity eigenvalues exceed those of the equilibrium. A comparison of these conditions with those of Frieman and Rotenberg is discussed.