We study two-dimensional steady incompressible Euler flows in an infinite strip under slip boundary conditions. A natural question concerns identifying geometric conditions under which such flows must be parallel shears. This question has attracted much attention only in the recent decade, during which many important results have emerged. While several rigidity results provide sufficient conditions, our aim is to establish a necessary and sufficient characterization that also allows for stagnation points. We show that a steady Euler flow is a parallel shear flow if and only if almost every regular streamline traverses the strip longitudinally. The proof relies on geometric estimates and analysis of the asymptotic behavior of solutions at far fields. This is joint work with my UM colleagues Changfeng Gui, Huan Xu, and Wen Yang.