Summary: We develop microstructural models to examine the mechanisms responsible for differences in tensile and compressive strength observed in cellular materials. We limit our analyses to those materials that fail by the same mechanism (yielding or microcracking) in tension and compression. Using both a honeycomb and an open-cell foam model, we demonstrate that density-dependent, compression-strong strength asymmetry arises when two conditions are met: the cell wall material has a higher yield strength in compression than in tension, and the cell walls are loaded simultaneously by axial forces and bending moments. Our models predict that strength asymmetry (defined as the ratio of compressive to tensile yield strength of the cellular material) inereases with relative density (as observed in real materials such as rigid polyurethane foams and trabecular bone), and that strength asymmetry is more pronounced in anisotropic materials (where oblique struts are more closely aligned with the direction of loading).