A novel magnetoelastic mechanical metamaterial consisting of a hyperelastic 2D lattice incorporating permanent magnets is presented and characterized. When properly designed and fabricated, the metamaterial possesses two stable equilibrium configurations (henceforth referred to as hexagonal/hourglass and kagome), both stretching dominated (and hence stiff ). The two configurations have significantly different elastic properties and wave propagation characteristics, as shown numerically and experimentally. By switching between the two configurations via uniaxial loading cycles, the material displays hysteresis, thus dissipating substantial amounts of energy; in contrast with purely mechanical bistable structures (e.g., arches, hinged beams and buckled beams), the proposed magnetoelastic metamaterial does not require multiple unit cells in series or stiff boundary conditions to exhibit energy dissipation, thus enabling the implementation of compact stiff dampers. The presence of a bandgap in the kagome configuration (but not in the hexagonal/hourglass configuration) is attributed to an internal resonance mechanism and provides the foundation for the development of compact dynamic filters for mechanical signals.