Folding flat sheets into three-dimensional origami systems has applications in various fields, ranging from large-scale reconfigurable structures to small-scale architected materials called metamaterials. One intriguing property of 2D origami metamaterials is that their out-of-plane bending performance conflicts with that of in-plane. Unlike plates made of conventional material, which show identical Poisson effects in bending and stretching tests, origami metamaterials show a reversal of Poisson effects when subjected to these two tests. Motivated by this exciting property, the present study aims to systematically design metamaterials with different values of Poisson's ratio in these two tests. Towards this, an energy-based homogenization framework is first developed to characterize the exotic mechanics of origami systems. To account for the unique mechanics of origami systems, the framework includes out-of-plane curvature fields in addition to the in-plane strain fields commonly considered in the homogenization of planar lattice structures. This results in a couple-stress continuum with second gradient fields for the homogeneous response of these lattices. The proposed framework is validated through analytical results and numerical simulations of finite lattices. Then, building on the homogenization framework, a topology optimization technique is proposed for the inverse design of origami-inspired metamaterials for chosen Poisson behaviour in stretching and bending. This study unveils 2D lattice-structured metamaterials exhibiting distinctive values of Poisson's ratio in stretching and bending tests, a possibility unknown earlier.