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We propose a magnetic field-induced asymmetric mechanical metamaterial. The tunable metamaterial integrates hard-Magnetic Active Elastomers (hMAEs) into the microstructural design combining snap-through, bi-stability, and local resonant effects. The proposed metamaterial design comprises resonating units made out of hMAE, and these units are supported by highly deformable curved beams connected with an elastomeric matrix. Activated by a magnetic field, the resonating units attain an unstable regime with dramatic configuration and stiffness variation. These controlled transformations significantly affect the elastic wave propagation. We illustrate that the proposed magnetoactive metamaterial enables bandgap tunability over a broadband low-frequency range. Thus, the proposed design allows remote and reversible control of the metamaterial performance. Moreover, the hMAE-based systems can incorporate the polarity and chirality stemming from the interaction of external magnetic fields and hMAE phases, giving rise to the unusual behavior of the metamaterial, and potentially enabling elastic cloaking.