The present paper shows that the usual factor analytic structured data dispersion matrix lambda psi lambda' + delta can readily arise from a set of scores y = lambda eta + epsilon, shere the "common" (eta) and "unique" (epsilon) factors have nonzero covariance: gamma = Cov epsilon,eta) is not equal to 0. Implications of this finding are discussed for the indeterminacy of factor scores, and for the issue of invariance of factor analytic covariance models. The size of the problem is explored with numerical examples.