We present a density-functional theory trend study addressing the binding of the trans-cis conformationalswitch azobenzene ͑C 6 H 5 -N= N-C 6 H 5 ͒ at three coinage-metal surfaces. From the reported detailed energetic-, geometric-, and electronic-structure data we conclude that the governing factor for the molecule-surface interaction is a competition between covalent bonding of the central azo ͑-N= N-͒ bridge on the one hand and the surface interaction of the two closed-shell phenyl ͑-C 6 H 5 ͒ rings on the other. With respect to this factor the cis conformer exhibits a more favorable gas-phase geometric structure and is thus more stabilized at the studied surfaces. With the overall binding still rather weak the relative stability of the two isomers is thereby reduced at Ag͑111͒ and Au͑111͒. This is significantly different at Cu͑111͒, where the cis bonding is strong enough to even reverse the gas-phase energetic order at the level of the employed semilocal electronic exchange and correlation ͑xc͒ functional. While this actual reversal may well be affected by the deficiencies due to the approximate xc treatment, we critically discuss that the rationalization of the general effect of the surface on the metastable molecular states is quite robust. This should equally hold for the presented analysis of recent tip-manipulation and photoexcitation isomerization experiments from the view point of the derived bonding mechanism.