Borsboom (Psychometrika, 71:425-440, 2006) noted that recent work on measurement invariance (MI) and predictive invariance (PI) has had little impact on the practice of measurement in psychology. To understand this contention, the definitions of MI and PI are reviewed, followed by results on the consistency between the two forms of invariance in the general case. The special parametric cases of factor analysis (strict factorial invariance) and linear regression analyses (strong regression invariance) are then described, along with findings on the inconsistency between the two forms of invariance in this context. Two numerical examples of inconsistency are reviewed in detail. The impact of violations of MI on accuracy of selection is illustrated. Finally, reasons for the slow dissemination of work on invariance are discussed, and the prospects for altering this situation are weighed.