Thirty eight–atom binary clusters composed of elements from groups 10 and 11 of the Periodic Table mixing a second-row with a third-row transition metal (TM) (i.e., clusters composed of the four pairs: Pd–Pt, Ag–Au, Pd–Au, and Ag–Pt) are studied through a combined empirical-potential (EP)/density functional (DF) method. A “system comparison” approach is adopted in order to analyze a wide diversity of structural motifs, and the energy competition among different structural motifs is studied at the DF level for these systems, mainly focusing on the composition 24-14 (the first number refers to the second-row TM atom) but also considering selected motifs with compositions 19-19 (of interest for investigating surface segregation effects) and 32-6 (also 14-24 and 6-32 for the Pd–Au pair). The results confirm the EP predictions about the stability of crystalline structures at this size for the Au–Pd pair but with decahedral or mixed fivefold-symmetric/closed-packed structures in close competition with fcc motifs for the Ag–Au or Ag–Pt and Pd–Pt pairs, respectively. Overall, the EP description is found to be reasonably accurate for the Pd–Pt and Au–Pd pairs, whereas it is less reliable for the Ag–Au and Ag–Pt pairs due to electronic structure (charge transfer or directionality) effects. The driving force to core-shell chemical ordering is put on a quantitative basis, and surface segregation of the most cohesive element into the core is confirmed, with the exception of the Ag–Au pair for which charge transfer effects favor the segregation of Au to the surface of the clusters.