Structural distortions in cuprate materials offer a microscopic origin for anisotropies in electron transport in the basal plane. Using a real-space Hartree-Fock approach, we consider the ground states of the anisotropic Hubbard (tx = ty) and t-J (tx = ty, Jx = Jy) models. Symmetrical but inhomogeneous ("polaronic") charge structures in the isotropic models are altered even by rather small anisotropies to one-dimensional, stripe-like features. We find two distinct types of stripe, namely uniformly filled, antiphase domain walls and non-uniform, half-filled, in-phase ones. We characterize their properties, energies and dependence on the model parameters, including filling and anisotropy in t (and J). We discuss the connections among these results, other theoretical studies and experimental observation.