Heterogeneous catalysis, which is widely used in the chemical industry, makes a great use of supported late-transition-metal nanoparticles, and bimetallic catalysts often show superior catalytic performances as compared to their single metal counterparts. In order to optimize catalyst efficiency and discover new active combinations, an atomic-level understanding and control of the catalyst structure is desirable. In this work, the structure of catalytically active AuRh bimetallic nanoparticles prepared by colloidal methods and immobilized on rutile titania nanorods was investigated using aberration-corrected scanning transmission electron microscopy. Depending on the applied post-treatment, different types of segregation behaviours were evidenced, ranging from Rh core -Au shell to Janus via Rh ball - Au cup configuration. The stability of these structures was predicted by performing density-functional-theory calculations on unsupported and titania-supported Au-Rh clusters; it can be rationalized from the lower surface and cohesion energies of Au with respect to Rh, and the preferential binding of Rh with the titania support. The bulk-immiscible AuRh/TiO2 system can serve as a model to understand similar supported nanoalloy systems and their synergistic behaviour in catalysis.