Hybrid organic-inorganic perovskites are famous for the diversity of their chemical compositions, phases, phase transitions, and associated physical properties. We use a combination of experimental and computational techniques to reveal a strong coupling between structure, magnetism, and spin splitting in a representative of the largest family of hybrid organic-inorganic perovskites: the formates. With the help of first-principles simulations, we find spin splitting in both conduction and valence bands of [NH2NH3]Co(HCOO)3 induced by spin-orbit interactions, which can reach up to 14 meV. Our magnetic measurements reveal that this material exhibits canted antiferromagnetism below 15.5 K. The direction of the associated antiferromagnetic order parameter is strongly coupled with spin splitting in the centrosymmetric phase, allowing for the creation and annihilation of spin splitting through the application of a magnetic field. Furthermore, the structural phase transition to the experimentally observed polar Pna21 phase completely changes the aforementioned spin splitting and its coupling to magnetic degrees of freedom. This reveals that in [NH2NH3]Co(HCOO)3, the structure and magnetism are strongly coupled to spin splitting and can be manipulated through electric and magnetic fields. We believe that our findings offer an important step toward a fundamental understanding and practical applications of materials with coupled properties.