Scalar conservation law $\partial_t \rho(t, x) + \partial_x(f (t, x, \rho)) = 0$ with a smooth flux in the state variable $\rho$ piecewise smooth in the $(t, x)$-plane admits infnitely many consistent notions of solution which differ by the choice of interface coupling. Only the case of the so-called vanishing viscosity solutions received full attention, while different choice of coupling is relevant in modeling situations that appear, e.g., in road traffic and in porous medium applications. In this paper, existence of solutions for a wide set of coupling conditions is established under some restrictions on $f$, via a finite volume approximation strategy adapted to slanted interfaces and to the presence of interface crossings. The notion of solution, restated under the form of an adapted entropy formulation which is consistently approximated by the numerical scheme, implies uniqueness and stability of solutions. Numerical simulations are presented to illustrate the reliability of the scheme.