Recent work with the 42 mental ability tests administered to participants of the Minnesota Study of Twins Reared Apart (MISTRA) has suggested that there are important dimensions of mental ability that function independently of "g". Two of these dimensions, rotation-verbal and focus-diffusion, appear to involve trade-offs: greater residual rotation ability implies less residual verbal ability and vice-versa, and the focus-diffusion dimension functions similarly. These two uncorrelated dimensions also show strong sex differences. Individuals lying at different positions along these dimensions may have brains that differ structurally and/or functionally, leading to differences in the ways they approach mental ability tasks. If so, we should expect a lack of factorial invariance in test scores across groups of individuals lying at different positions on the two dimensions, indicating that the tests do not measure the same constructs in the same ways in the different groups. This study demonstrated such a lack of factorial invariance in each of the three mental ability batteries included in the MISTRA assessment. The Wechsler Adult Intelligence Scale can be scored to correlate 0.7 with each of the two dimensions. We propose that use of these scores may help to clarify brain-mapping studies relating brain structure and function and to facilitate understanding of sex differences in mental abilities.