Density functional theory calculations were performed on both free and TiO2(110)- supported 38-atom truncated octahedron Au-Rh nanoalloy clusters at various compositions to investigate Au-Rh alloying properties at the nanoscale and the influence of the support. Furthermore, H2, O2, and CO adsorption properties were also investigated with a view to estimating and optimizing the catalytic performance of Au-Rh nanoalloy systems. Among the structures studied, the Rh core-Au shell morphology was found to be the lowest energy structure for bare clusters, while molecular adsorption energies are greatest on surface Rh atoms. It is also shown that multiple CO and O2 adsorption can change the stability order of nanoalloy clusters. For example upon adsorption of 6 molecules of CO or O2 the centroid structure, which has 6 surface Rh atoms, becomes the lowest energy structure for the Au32Rh6 composition.