Theoretical analysis and experimental results are presented to demonstrate the universal characteristics of symmetry-breaking bifurcations for pattern-forming systems in a circular domain. A comparison is made between stationary and rotating patterns of concentric rings of cells from experiments on premixed flames. Cells belonging to stationary rings are symmetric, while those of rotating rings are not. These results are reproduced in a model. Normal form equations for the Fourier-Bessel coefficients are deduced, and are seen to be those which lead to parity-breaking.