The relative stabilities of different chemical arrangements of Pd-Ir and Au-Rh nanoalloys (and their pure metal equivalents) are studied, for a range of compositions, for fcc truncated octahedral 38- and 79-atom nanoparticles (NPs). For the 38-atom NPs, comparisons are made of pure and alloy NPs supported on a TiO2(110) slab. The relative energies of different chemical arrangements are found to be similar for Pd-Ir and Au-Rh nanoalloys, and depend on the cohesive and surface energies of the component metals. For supported nanoalloys on TiO2, the interaction with the surface is greater for Ir (Rh) than Pd (Au): most of the pure NPs and nanoalloys preferentially bind to the TiO2 surface in an edge-on configuration. When Au-Rh nanoalloys are bound to the surface through Au, the surface binding strength is lower than for the pure Au NP, while the Pd-surface interaction is found to be greater for Pd-Ir nanoalloys than for the pure Pd NP. However, alloying leads to very little difference in Ir-surface and Rh-surface binding strength. Comparing the relative stabilities of the TiO2-supported NPs, the results for Pd-Ir and Au-Rh nanoalloys are the same: supported Janus NPs, whose Ir (Rh) atoms bind to the TiO2 surface, bind most strongly to the surface, becoming closer in energy to the core-shell configurations (Ir@Pd and Rh@Au) which are favoured for the free particles.