3-component particle image velocitmetry data from a model swirl combustor are analyzed. Our previous work focused on the spontaneous transition from the stable state to the thermoacoustically excited state. Time-frequency (bandpass filterering) methods revealed a sequence of quasi-periodic modes, intermittent in time and in space. In this paper, we focus on the coexistence and interactions between two modes in the stable regime. The simplest is the axisymmetric mode, with vortex rings as the organizing flow structure. More complicated is the precessing mode, which combines a pair of counter-rotating helical vortices and a precessing recirculating vortex core. Using dynamic mode decomposition (DMD) yields different frequencies for the dominant modes. We reconcile the two methods by noting that DMD yields the fastest growing modes, whereas Fourier-related methods yield energy-dominant modes. We show that, on average, the growth of the helical or the axisymmetric mode precedes its peak energy by one to four periods. We also show that the mode amplitudes are correlated with varying signs across the burner.