The mechanics of cellular honeycombs-part of the rapidly growing field of architected materials-in addition to its importance for engineering applications has a great theoretical interest due to the complex bifurcation mechanisms leading to failure in these nonlinear structures of high initial symmetry. Of particular interest to this work are the deformation patterns and their stability of finitely strained circular cell honeycomb. Given the high degree of symmetry of these structures, the introduction of numerical imperfections is inadequate for the study of their behavior past the onset of first bifurcation. Thus, we further develop and explain a group-theoretic approach to investigate their deformation patterns, a consistent and general methodology that systematically finds bifurcated equilibrium orbits and their stability. We consider two different geometric arrangements, hexagonal and square, biaxial compression along loading paths, either aligned or at an angle with respect to the axes of orthotropy, and different constitutive laws for the cell walls which can undergo arbitrarily large rotations, as required by the finite macroscopic strains applied. We find that the first bifurcation in biaxially loaded hexagonal honeycombs of infinite extent always corresponds to a local mode, which is then followed to find the deformation pattern and its stability. Depending on load path orientation, these first bifurcations can be simple, double or even triple. All bifurcated orbits found are unstable and have a maximum load close to their point of emergence. In contrast, the corresponding instability in square honeycombs always corresponds to a global mode and hence the deformation pattern will depend on specimen size and boundary conditions.