Many closed-cell foams exhibit an elongated cell shape in the foam rise direction, resulting in anisotropic compressive properties, e.g. modulus and strength. Nevertheless, the underlying deformation mechanisms and how cell shape anisotropy induces this mechanical anisotropy are not yet fully understood, in particular for the foams with a high cell face fraction and low relative density. Moreover, the impacts of mesostructural stochastics are often overlooked. This contribution conducts a systematic numerical study on the anisotropic compressive behaviour of low-density closed-cell foams (with a relative density <0.15), which accounts for cell shape anisotropy, cell structure and different mesostructural stochastics. Representative volume elements (RVE) of foam mesostructures are modelled, with cell walls described as Reissner–Mindlin shells in a finite rotation setting. A mixed stress–strain driven homogenization scheme is introduced, which allows for enforcing an overall uniaxial stress state. Uniaxial compressive loadings in different global directions are applied. Quantitative analysis of the cell wall deformation behaviour confirms the dominant role of membrane deformation in the initial elastic region, while the bending contribution gets important only after buckling, followed by membrane yielding. Based on the identified deformation mechanisms, analytical models are developed that relate mechanical anisotropy to cell shape anisotropy. It is found that cell shape anisotropy translates into the anisotropy of compressive properties through three pathways, cell load-bearing area fraction, cell wall buckling strength and cell wall inclination angle. Besides, the resulting mechanical anisotropy is strongly affected by the cell shape anisotropy stochastics while almost insensitive to the cell size and cell wall thickness stochastics. The present findings provide deeper insights into the relationships between the anisotropic compressive properties and mesostructures of low-density closed-cell foams.