Greigite (Fe3S4) and magnetite (Fe3O4) are isostructural and isoelectronic ferrimagnets with quite distinct properties. Electronic structure calculations reveal greigite is a normalmetal in contrast to half-metallic magnetite. Greigite shows a complex Fermi surface with a unique influence of relativistic effects: The existence of sheets of the Fermi surface depends on the direction of the magnetization. This enables spinorbitronics, spintronics on the level of a single compound rather than a device. Due to its relativistic origin, spin contamination is irrelevant in spinorbitronics and the entire periodic table is available for optimizations.