This article shows how interfactor correlation is affected by error correlations. Theoretical and practical justifications for error correlations are given, and a new equivalence class of models is presented to explain the relationship between interfactor correlation and error correlations. The class allows simple, parsimonious modeling of error correlations via prespecifying reliabilities. Within the class, the correlation between latent factors can be as high as 1.0, and as low as the correlation between certain component scores. The models are indistinguishable in terms of parameter parsimony, identifiability, and fit statistics, implying that interfactor correlation is not identifiable within the class. The existence of the class is problematic for psychometric measurement, because estimates of interfactor correlation form the foundation of much of the literature. (Contains 4 figures.)