The structures of AgCu clusters containing 40 atoms are investigated. The most promising structural families (fcc clusters, capped decahedra, and two types of capped polyicosahedra) are singled out by means of global optimization techniques within an atom-atom potential model. Then, representative clusters of each family are relaxed by means of density-functional methods. It is shown that, for a large majority of compositions, a complex interplay of geometric and electronic shell-closure effects stabilizes a specific polyicosahedral family, whose clusters are much lower in energy and present large HOMO-LUMO gaps. Within this family, geometric and quantum effects concur to favor magic structures associated with core-shell chemical ordering and high symmetry, so that these clusters are very promising from the point of view of their optical properties. Our results also suggest a natural growth pathway of AgCu clusters through high-stability polyicosahedral structures. Results for AuCu clusters of the same size are reported for comparison, showing that the interplay of the different effects is highly material specific.