In this paper we investigate the effects that breaking the symmetry in an annular configuration has on the features of standing, spinning and slanted limit cycle oscillations of thermoacoustic instabilities coupled by azimuthal modes. Experiments are carried out in the MICCA combustor, an annular system comprising sixteen identical matrix injectors. The symmetry of the system is broken by partially blocking the injectors or by changing their geometry. The resulting pressure fields are analysed using a newly introduced pressure ansatz which serves to characterize the nature of the instability (standing or spinning) and the pressure antinodes. Two distinct scenarios are identified. First, when the distribution of the unsteady rate of heat release is slightly changed, for example by partially blocking a single injector, the nature of the limit cycle remains the same. A spinning/standing mode in the symmetric combustor remains spinning/standing in the combustor in which the symmetry is weakly broken. However, it is also found that the orientation of the pressure antinodal line does change and that it is controlled by the pattern of asymmetries introduced in the system. Second, when the distribution of unsteady rate of heat release is changed to a larger extent, for example by partially blocking several burners and by changing their geometries, the nature of the limit cycle can change. A dynamical system based on the pressure ansatz is used to gain insight into the experimental observations. Using a model flame describing function it is shown that one can retrieve many of the features observed in the experimental data, including the nature of the instabilities and the pressure antinodal position as a function of the injection asymmetries. The dynamical system is also used to assess the strength of inhomogeneity introduced in the system. As in the experiments, it is found that in the model starting from a symmetric combustor and then partially blocking a single burner (by modifying its FDF) one is able to alter the pressure antinode. However if in the model one starts from a system with a strong variation of the unsteady heat release rate in the azimuthal direction, partially blocking one injector does not change perceptively any of the features of the predetermined limit cycle.