It is pointed out that the classical rules for matching at a tangential discontinuity, viz., continuity of the pressure and displacement, are nonuniversal. It is shown that in open systems both the pressure and the displacement can be discontinuous. The presence of a jump in the displacement renders meaningless the textbook use of the hydrodynamic concept of a perturbed surface of tangential discontinuity: the perturbed tangential discontinuity is not a single surface but a small spatial region bounded by nonparallel surfaces. It turns out that matching rules which are independent of the structure of the jump can be obtained only for a narrow (one-parameter) class of tangential discontinuities. In general the matching rules are different for different structures of the jumps in the main parameters of the medium. The examples cited in this paper are based on the real tangential discontinuity observed in the gaseous disk of the Milky Way galaxy.