Density functional theory calculations were performed on both free and TiO2(110)-supported 38- and 79 atom truncated octahedral Au-Rh nanoalloy clusters of 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 RhcoreAushell morphology was found to be the lowest energy structure for bare clusters- while molecular adsorption energies are greatest on surface Rh atoms. It is shown that multiple CO and O2 adsorption can change the stability order of nanoalloy clusters selectively. For example upon adsorption of 6 molecules of CO (or O2) the centroid (or hex) structures- which have 6 surface Rh atoms- become the lowest energy structures for the Au32Rh6 composition. It is also shown that Janus type clusters become more competitive with the RhcoreAushell on the TiO2(110)-support due to the surface induced stabilisation. Adsorption properties of CO and O2 were found to be similar on TiO2(110)-supported Au-Rh nanoalloys: molecular adsorption is favoured on Rh atoms- while on Au sites O2 molecules prefer to bridge Au and a surface Ti atom.