The selective excitation of localized surface wave modes remains a challenge in the design of both leaky-wave and bound-wave devices. In this Letter, we show how the truncation of a metasurface can play an important role in breaking the spatial inversion symmetry in the excitation of surface waves supported by the structure. This is done by combining a large anisotropy in the dispersion relation and the presence of an edge that also serves as a coupling mechanism between the plane wave excitation and the induced surface waves. By resorting to the exact solution to the scattering problem based on a discrete Wiener–Hopf technique, we show that by inverting the component of the impinging wavevector parallel to the truncation, two distinct surface waves are excited.