Accepted theory for anisotropic flow in porous media establishes that the properties of a particular flow may depend upon the flow orientation, but generally assumes that flow properties are invariant for a reversal of the flow direction. By simulating simple two-dimensional and three-dimensional flows from the pore-scale, we demonstrate that while this assumption holds true when flow is slow such that the approximations supporting Darcy's law apply, reversal of the flow direction can have a significant impact on nonlinear corrections to Darcy's law that become important at higher flow rates. In this study, we consider flow through simple periodic porous media consisting of oriented, asymmetrical grains for Reynolds numbers < 150. Analysis of the pore-scale flow structure demonstrates that direction-dependent effects can be linked with asymmetry. We present a nonlinear correction to Darcy's law that accounts for this extended anisotropy and propose a macroscopic morphological measure to quantify asymmetry of the solid phase.