This work addresses the fundamental difference in behavior between magnetorheological elastomers (MREs) formed from soft-magnetic particles, whose behavior is driven by local demagnetizing effects and those formed with hard-magnetic particles that have a preferred magnetic axis and therefore generate magnetic torques at the particle level. This work explores the phenomena by defining and examining four classes of MREs based upon permutations of particle alignment - magnetization pairs, i.e. I-I for magnetically isotropic particles arranged isotropically (randomly, or unaligned), A-A for magnetically anisotropic particles arranged anisotropically (typically aligned in chains), etc. The distinctions are important since the particle-field interactions for each class differ substantially. The behavior of classes I-I and I-A are driven primarily by demagnetizing effects while classes I-A and A-A are driven by the torques produced in the particles. MRE materials made with barium hexaferrite (BaM) (Classes A-A and I-A) and Fe powders (Classes A-I and I-I), aligned and unaligned, served as proxies for each of the four classes in this work. BaM, with saturation magnetization Msat = 4 x 10 5 A/m and coercive field Hc > 3 x 105 A/m, provided the magnetically anisotropic behavior while iron, with Msat =1.8 x 106 A/m and Hc < 2 x 103 A/m, provided the soft magnetic behavior. Experiments on materials with 30% v/v particle concentrations showed that under uniform magnetic fields class A-A (aligned BaM) MREs were capable of large deflections in cantilever beam bending (deflections of 12mm for length 50mm and magnetic field 1.2 x 105 A/m) whereas all other classes, including I-A (random BaM) MREs, showed none. Tip deflection varied linearly with applied field strength. Tip blocking-force versus deflection experiments were also conducted on cantilevered A-A specimens. These tests showed that tip force increased with decreasing free deflection and with increasing field strength.