Step edges represent a local break of lateral symmetry in ultrathin magnetic films. In our experiments, we investigate the spin coupling across atomic step edges on Fe/W(110) by means of spin-polarized scanning tunneling microscopy and spectroscopy. Local modifications of the spin texture toward step edges separating double from single layer areas are observed, and selection rules indicate a chiral spin coupling that significantly changes with the propagation along the [110] or the [001] crystallographic direction. The findings are explained via anisotropic Dzyaloshinskii-Moriya interactions arising from the broken lateral symmetry at atomic step edges.