A numerical investigation of the onset of three-dimensional states in Taylor–Couette flow for aspect ratio one is presented. Two main branches exist, one preserving and the other breaking the reflection symmetry about the mid-plane. Both branches become three-dimensional via Hopf bifurcations to rotating waves with different azimuthal wavenumbers. Moreover, the symmetric branch exhibits secondary Hopf bifurcations and transitions to complex spatio-temporal dynamics at Reynolds numbers $\hbox{\it Re}\,{\sim}\,1000$. The analysis of the three-dimensional solutions shows that the dynamics is driven by the jet of angular momentum erupting from the inner cylinder boundary layer and its interactions with the sidewall and endwall layers. The various solutions are organized by a lattice of spatial and spatio-temporal symmetry subgroups which provides a framework for the relationships between the solution types and for the symmetry-breaking bifurcations. The results obtained agree with previous experimental results and help clarify many aspects of the mode competition at the higher $\hbox{\it Re}$ values.