Low-frequency combustion instabilities, and ultimately, engine-level dynamics in can-annular gas turbine combustion systems, are driven by acoustic interactions between adjacent combustors through an annular cross-talk section upstream of first stage turbine nozzles. This is an area of substantial technological challenges, because experimental and numerical complexities scale exponentially with the number of combustors, restricting most investigations to a single combustor or a coupled two-combustor configuration. We recently reported that in a can-annular combustion system consisting of four identical lean-premixed combustors pressure oscillations occur in many combinations of characteristic patterns and frequencies, including symmetric (push-push) and asymmetric (push-pull) interactions, when the system is subjected to rotationally symmetric operating conditions. The impact of rotational asymmetry on can-annular thermoacoustics is still a largely open subject; here we use the same experimental setup to perform multiple independent analyses of the effect of broken rotational symmetry on large-scale pattern formations and modal dynamics of multiple eigenmodes. We demonstrate that the presence of rotational asymmetry can lead to fundamentally different dynamic states, including pairwise push-pull modes, spinning azimuthal instabilities, superposition modes, and strong mode localization, which are totally absent under rotationally symmetric conditions. Of particular importance is the experimental observation of spinning azimuthal instabilities in the annular cross-talk section, which are acoustically separate from flame-dynamics-driven standing wave motion in the flame tube sections. This leads to the coexistence of traveling and standing waves in the can-annular system. In conjunction with FEM-based Helmholtz simulations, we determine the origin of the azimuthal modes to be the simultaneous excitation of two different pairwise push-pull interaction modes at the same frequency - a phenomenon known as degenerate eigenmodes.