The buckling and crushing mechanics of cellular honeycomb materials is an important engineering problem. Motivated by the pioneering experimental and numerical studies of Papka and Kyriakides (1994, 1999a,b), we review the literature on finitely strained honeycombs subjected to in-plane loading and identify two open questions: (i) How does the mechanical response of the honeycomb depend on the applied loading device? and (ii) What can the Bloch wave representation of all bounded perturbations contribute to our understanding of the stability of post-bifurcated equilibrium configurations? To address these issues we model the honeycomb as a two-dimensional infinite perfect periodic medium. We use analytical group theory methods (as opposed to the more common, but less robust, imperfection method) to study the honeycomb's bifurcation behavior under three different far-field loadings that produce (initially) the same equi-biaxial contractive dilatation. Using an FEM discretization of the honeycomb walls (struts), we solve the equilibrium equations to find the principal and bifurcated equilibrium paths for each of the three loading cases. We evaluate the structure's stability using two criteria: rank-one convex-ity of the homogenized continuum (long wavelength perturbations) and Bloch wave stability (bounded perturbations of arbitrary wavelength). We find that the post-bifurcation behavior is extremely sensitive to the applied loading device, in spite of a common principal solution. We confirm that the flower mode is always unstable, as previously reported. However, our (first ever) Bloch wave stability analysis of the post-bifurcated equilibrium paths shows that the flower mode is stable for all sufficiently short wavelength perturbations. This new result provides a realistic explanation for why this mode has been observed in the finite size specimen experiments of Papka and Kyriakides (1999a).