We study ${\Bbb Z}^2$ actions arising as base point actions of tiling representations of ${\Bbb R}^2$ flows. We cast an equivalence relation between such actions in terms of a simple arithmetic condition on an orbit equivalence. Stated as such, our equivalence class is easily seen to be a restricted even Kakutani equivalence, as we well as a higher-dimensional generalization of $\alpha$-equivalence, defined by Fieldsteel, del Junco and Rudolph for ${\Bbb Z}$ actions.