The structure of Pd clusters adsorbed on MgO(001) is determined by a combination of global-optimization methods using semiempirical potentials and density functional calculations. The transition to fcc clusters with (001) epitaxy is shown to take place in the size range 10<or=N<or=15 atoms. Truncations of vertices and basal corners in fcc epitaxial clusters proceed in parallel, while overhangs are already formed at small sizes, below 30 atoms.