Antiferromagnetic insulators are a ubiquitous class of magnetic materials, holding the promise of low-dissipation spin-based computing devices that can display ultra-fast switching and are robust against stray fields. However, their imperviousness to magnetic fields also makes them difficult to control in a reversible and scalable manner. Here we demonstrate a novel proof-of-principle ionic approach to control the spin reorientation (Morin) transition reversibly in the common antiferromagnetic insulator α-Fe₂O₃ (haematite) - now an emerging spintronic material that hosts topological antiferromagnetic spin-textures and long magnon-diffusion lengths. We use a low-temperature catalytic-spillover process involving the post-growth incorporation or removal of hydrogen from α-Fe₂O₃ thin films. Hydrogenation drives pronounced changes in its magnetic anisotropy, Néel vector orientation and canted magnetism via electron injection and local distortions. We explain these effects with a detailed magnetic anisotropy model and first-principles calculations. Tailoring our work for future applications, we demonstrate reversible control of the room-temperature spin-state by doping/expelling hydrogen in Rh-substituted α-Fe₂O₃.