We consider the edge Hall conductance and show it is invariant under perturbations located in a strip along the edge. This enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the right half plane. As a particular interesting case we put forward a general quantization formula for the difference of edge Hall conductances in semi-infinite samples with and without a confining wall. Applications to disordered Hall systems with and without a confining potential are discussed.