Solitons are large-amplitude, spatially confined wave packets in nonlinear media. They occur in a wide range of physical systems, such as water surfaces, optical fibres, plasmas, Bose-Einstein condensates and magnetically ordered media(1,2). A distinguishing feature of soliton behaviour that is common to all systems, is that they propagate without a change in shape owing to the stabilizing effect of the particular nonlinearity involved(1,3). When the propagation path is closed, modes consisting of one or several solitons may rotate around the ring, the topology of which imposes additional constraints on their allowed frequencies and phases(4,5). Here we measure the mode spectrum of spin-wave solitons in a nonlinear active ring constructed from a magnetic ferrite film. Several unusual symmetry-breaking soliton-like modes are found, such as 'Mobius' solitons, which break the fundamental symmetry of 2pi-periodicity in the phase change acquired per loop: a Mobius soliton needs to travel twice around the ring to meet the initial phase condition.