A group-theoretical framework to describe vacancy ordering and magnetism in the Fe1−xS system is developed. This framework is used to determine the sequence of crystal structures consistent with the observed magnetic structures of troilite (FeS), and to determine the crystallographic nature of the low-temperature Besnus transition in Fe0.875S. It is concluded that the Besnus transition is a magnetically driven transition characterized by the rotation of the moments out of the crystallographic plane to which they are confined above the transition, accompanied by small atomic displacements that lower the symmetry from monoclinic to triclinic at low temperatures. Based on the phase diagram, magnetically driven phase transitions at low temperatures are predicted in all the commensurate superstructures of pyrrhotite. Based on the phase diagram, magnetically driven spin reorientations at low temperatures are predicted in all the commensurate superstructures of pyrrhotite. The exact nature of the spin rotation is determined by the symmetry of the vacancy-ordered state and based on this spin-flop transitions in 3C and 5C pyrrhotite and a continuous rotation akin to that seen in 4C pyrrhotite are predicted. A Besnus-type transition is also possible in 6C pyrrhotite. Furthermore, it is clarified that 3C and 4C pyrrhotite carry a ferrimagnetic moment whereas 5C and 6C are antiferromagnetic.