The consequences of conservation of angular momentum in, single- or double-polytropic, steady, compressible, magnetohydrodynamic (MHD) flow parallel to the magnetic field are examined, the principal result being the MHD counterparts of Crocco's theorem and Lord Kelvin's theorem, both expressed in terms of a generalized vorticity. Under special assumptions concerning geometry and/or homogeneity of the general field line invariants, these vortex laws lead to additional field line invariants which describe the conservation of generalized angular momentum in the plasma, taking into account torques produced by j¿B forces and Coriolis forces. ¿ American Geophysical Union 1994