Scanning tunneling microscopy and spectroscopy reveals a distinct asymmetry in the scattering properties of an isolated step for the n=1 image-potential state on Cu(001). The elastic scattering probability for an electron traveling downstairs is approximately two to three times higher than for the opposite upstairs direction. A one-dimensional scattering model is extended to the case of asymmetric transmission and reflection coefficients. The calculations explain the dispersion and the decay rate of the n=1 band on Cu(119) measured by two-photon photoemission. In particular the asymmetry of the decay rate can be described quantitatively with a minimum of adjustable parameters. While the results can be transferred successfully also to a Cu(1 1 15) surface, the limit of applicability is reached for a Cu(117) surface with a step separation of 3.5 nearest-neighbor distances.