Propagation of hydromagnetic surface waves on an MHD tangential discontinuity, separating two uniform magnetic plasmas, across which the magnetic field changes both its direction and field-strength, is examined. The study is arbitrarily confined to discontinuities one side of which has a plasma-β value larger than unity. A classification scheme to identify the ‘fast’ and the ‘slow’ surface waves is employed thus enabling us to find new features in the modal structure of these waves. In the case of sufficiently small values for the density ratio between the higher and the lower Alfvén speed regions on opposite sides of the interface, this study reveals the presence of a ‘forbidden region’ for propagation of fast surface waves with values of their propagation angle being smaller than a certain critical value. The study also reveals the existence of two distinct slow surface waves with angles of propagation larger than a certain critical value but for two different ranges of values for the angle between magnetic field vectors across the interface. While revealing such new structures of the surface modes for intermediate parameter values, the present investigation also confirms earlier results regarding the asymptotic behaviour of the modes in the incompressible limit.