The key issue of this contribution is the magneto-mechanical behavior of a magnetically hard magnetorheological elastomer (H-MRE). A numerical model based on the conservation laws of continuum mechanics and magnetism is presented and then applied to a specific 2-dimensional case of an H-MRE specimen. The interaction between two remanently magnetized particles and its dependence on the relative positioning of the particles to one another are studied. The reaction of the elastomer matrix to the particle-particle interaction is analyzed for three configurations of the particles in the specimen. In a second step, the specimen is exposed to a unidirectional tensile load. It is demonstrated that the magnetization of the particles induces a change in the modulus which strongly depends on the arrangement of the particles in the matrix: an increase as well as a decrease of the modulus are observed. By the help of the principal stresses, three different types of magnetic interactions are identified and assigned to the corresponding softening and stiffening reactions of the specimen.