Hybridization-related modifications of the first metal layer of a metal-organic interface are difficult to access experimentally and have been largely neglected so far. Here, we study the influence of specific chemical bonds (as formed by the organic molecules CuPc and PTCDA) on a Pb-Ag surface alloy. We find that delocalized van der Waals or weak chemical π-type bonds are not strong enough to alter the alloy, while localized σ-type bonds lead to a vertical displacement of the Pb surface atoms and to changes in the alloy's surface band structure. Our results provide an exciting platform for tuning the Rashba-type spin texture of surface alloys using organic molecules.