The trends in the bonding mechanism of 3,4,9,10-perylenetetracarboxylic acid dianhydride (PTCDA) to the Ag(111), Ag(100), and Ag(110) surfaces were analyzed on the basis of data obtained from x-ray standing waves and dispersion-corrected density functional theory. Of importance are the attractive local O-Ag bonds on the anhydride groups. They are the shorter, the more open the surface is, and lead even to partly repulsive interactions between the perylene core and the surface. In parallel, there is an increasing charge donation from the Ag surface into the π system of the PTCDA. This synergism explains the out-of-plane distortion of the adsorbed PTCDA and the surface buckling.