This paper deals with strategies of congruence studies, aimed at evaluating recoverableness of a given set of components from a first population in a second population where the same variables have been used. Five decisions inherent to congruence studies are analysed in detail. Confirmatory evidence with respect to recoverableness can be obtained from an independent component analysis for the second population, parallel to that of the first population. Disconfirmatory evidence requires oblique rotation to perfect congruence, which can always be attained. Rotation to perfect congruence is advocated as a new strategy, in which amounts of variance explained are of major concern. The perfect congruence strategy can be applied to variable-component correlations and to weights. The latter approach is to be preferred for two reasons. First, rotating weights to perfect congruence can be easily understood as a cross-validation method, closely related to the well-known multiple group method. Second, this approach appears to give more satisfactory results in practical applications than are obtained from rotating variable-component correlations to perfect congruence.