Recently, an Eulerian formulation based on evolving microstructural vectors was used (Lee and Rubin, 2020) to model anisotropic inelastic deformation rate in sheet metals. In contrast to Lagrangian formulations, the Eulerian formulation is insensitive to arbitrariness of reference and intermediate configurations. Here, additional constitutive equations are introduced to model asymmetric inelastic behaviors, such as the Bauchinger effect, permanent softening and the strength-differential effect within the context of the Eulerian formulation. The resulting model is calibrated to produce good agreement with several material data sets. In addition, example data sets are used to demonstrate advantages of the new model, especially for severe anisotropic response and large deformations. •An Eulerian formulation based on microstructural vectors is used for sheet metal.•Microstructural vectors determine elastic deformations and orthotropic axes.•Strength-differential Effect modeled using the Lode angle and a hardening parameter.•Baushinger effect and permanent softening in cyclic loading modeled using a directional hardening tensor.•The constitutive equations are validated using five material data sets.