Voltage-gated ion transport as a means of manipulating magnetism electrically could enable ultralow-power memory, logic and sensor technologies. Earlier work made use of electric-field-driven O2- displacement to modulate magnetism in thin films by controlling interfacial or bulk oxidation states. However, elevated temperatures are required and chemical and structural changes lead to irreversibility and device degradation. Here we show reversible and non-destructive toggling of magnetic anisotropy at room temperature using a small gate voltage through H+ pumping in all-solid-state heterostructures. We achieve 90° magnetization switching by H+ insertion at a Co/GdOx interface, with no degradation in magnetic properties after >2,000 cycles. We then demonstrate reversible anisotropy gating by hydrogen loading in Pd/Co/Pd heterostructures, making metal-metal interfaces susceptible to voltage control. The hydrogen storage metals Pd and Pt are high spin-orbit coupling materials commonly used to generate perpendicular magnetic anisotropy, Dzyaloshinskii-Moriya interaction, and spin-orbit torques in ferromagnet/heavy-metal heterostructures. Thus, our work provides a platform for voltage-controlled spin-orbitronics.