It is known that the profinite completions of a free semigroup which are associated with a pseudovariety of semigroups or of ordered semigroups, can be defined by an écart or a quasi-écart. We characterize those quasi-écarts and those quasi-uniformities which arise in this fashion. We also prove an Eilenberg-like one-to-one correspondence between pseudovarieties of ordered semigroups and so-called varieties of quasi-uniformities.