Thermoacoustic instabilities in annular gas turbine combustors often involve modes which vary in both the longitudinal and circumferential directions. Recent experimental studies show that during limit cycle oscillations, different thermoacoustic modes may be uncoupled, as is the case in purely longitudinal or circumferential spinning modes. They may also be coupled, for example two counter-rotating circumferential modes combining to give standing or mixed modes, and coupling between circumferential and longitudinal modes giving rise to the slanted mode. Accurately predicting such modal couplings and the resulting spatial pattern of limit cycle oscillations remains an open challenge. This work uses a 2-D low-order network model based on modal expansions, validated against a full 3-D Helmholtz solver, to systematically investigate these couplings. For the first time, low-order network modelling is shown to capture limit cycle oscillations exhibiting both uncoupled and nonlinearly coupled modes, the latter including coupling between counter-rotating circumferential modes and between longitudinal and circumferential modes. It is shown that limit cycle solutions with totally different mode patterns (longitudinal, circumferential spinning, circumferential standing and slanted) can all exist in a given thermoacoustic system, with switches between modal patterns arising from slight changes in parameters such as the flame time delay.