We investigate and elucidate the surprising observation of atomically ordered domains in dome-shaped SiGe nanoislands. We show, through atomistic Monte Carlo simulations, that this ordering is a surface-related phenomenon, and that is driven by surface equilibrium rather than by surface kinetics. The ordering depends on facet orientation. The main source of ordering is the {15 3 23} facet, while the {105} and {113} facets contribute less. Subsurface ordered configurations self-organize under this facet and are frozen-in and buried during island growth, giving rise to the ordered domains. Ordering mechanisms based on constrained surface kinetics, requiring step-mediated segregation at the island facets, are shown to be much less likely.