The author defines a \(\wedge\)-right congruence on a completely regular semigroup S as any left congruence \(\rho\) on S for which (i) \(a\rho\) \(\hat a\) and (ii) \(a\rho\) b implies \(\hat ac\rho\) \^bc for any a,b,c in S. (Here \(\hat a\) denotes the unit of \(H_ a)\). The dual notion defines \(\wedge\)-left congruences. Whether Green's relations are \(\wedge\)-left or \(\wedge\)-right congruences, or neither, is shown to determine various classes of orthodox completely regular semigroups.