Magnetorheological elastomers (MREs) are an emerging class of smart materials whose mechanical behavior varies in the presence of a magnetic field. Historically MREs have been comprised of soft-magnetic iron particles in a compliant matrix such as silicone elastomer. Numerous works have experimentally cataloged the MRE effect, or increase in shear stiffness, versus the applied field. Several other researchers have derived constitutive models for the large deformation behavior of MREs. In almost all cases the arrays of embedded particles, and or the particles themselves, are assumed magnetically symmetric with respect to the external magnetic field, i.e. the bulk materials exhibit magnetic symmetry in the given experimental or analytical configuration. In this work the author presents results of dynamic shear experiments, Lagrangian dynamic analysis, and static shear simulations on MRE material systems that exhibit broken magnetic symmetry. These new materials utilize barium hexaferrite powder as the magnetically anisotropic filler combined with a compliant silicone elastomer matrix. Simulations of representative laminate structures comprised of varied arrays of magnetic particles exhibit novel actuation behaviors including reversible shearing deformation, variable magnetostriction, and most surprisingly, piezomagnetism. Results of dynamic shear experiments and analytical modeling support predicted shearing actuation responses in MREs having broken symmetry and only in those systems.