We analyse the bifurcations of a general ordinary dififerential equation where is equivariant under an action of the group O(2) on. The equation represents the most general nonlinear local interaction of three O(2)-symmetric modes:a steady-state mode with mode-number k, and two periodic (Hopf) modes with mode-numbers l and m. The parameter λ is a bifurcation parameter, and α₁, α₂are unfolding parameters that split the individual modes apart. The system is assumed to be in Birkhoff normal form, so that f also commutes with an action of the 2-torus T². We discuss the existence and stability of bifurcating branches and how these break the O(2) × T² symmetry.Depending on the precise mode-numbers k l m we find up to 31 symmetry classes of possible solutions including six that combine all three modes, and thus cannot be found in any 2-mode interaction. We also discuss the possible occurrence of Sacher-Naimark torus bifurcations, providing a further 10 solution types, and 'slow drift'bifurcations. This 10-dimensional system can occur generically in O(2)-symmetric bifurcation problems having two extra parameters, and in principle is applicable to a wide range of physical systems. The discussion here is motivated by the observed pattern formation in the Taylor–Couette system, the flow of a fluid contained between coaxial rotating cylinders. It arises by seeking a 'hidden organizing centre' that combines two previous mode-interaction models of this system: a 6-dimensional Hopf–steady-state model due to Chossat and looss (1985) and Golubitsky and Stewart (1986), and an 8-dimensional Hopf–Hopf model due to Chossat, Demay and looss (1987). We interpret the general results on the 10-dimensional system in the context of Taylor–Couette flow, giving schematic pictures of the associated flow patterns. The model incorporates almost all of the observed non-chaotic flows in the Taylor–Couette experiment into a single finite-dimensional dynamical system. It predicts the possible occur- rence of four new flow patterns (corresponding to four of the six possible solutions that combine all three modes). Theseform invariant 3-tori, and may be described as superimposed twisted vortices, superimposed wavy vortices, and two types of twisted wavy vortices. Possible torus bifurcations from states in the 10-dimensional model include various modulated spirals, three types of modulated twisted vortices, three types of modulated wavy vortices, modulated superimposed spirals, modulated interpenetrating spirals, modulated superimposed ribbons, and modulated interpenetrating ribbons. However, whether any of these new states and torus bifurcations can actually occur in Taylor–Couette flow at suitable parameter values, and if so whether they can occur stably, depend upon more detailed numerical calculations than we have performed