Revell & Collar and eigenvectors

Revell & Collar (2009) show results from "tests of shared eigenstructure between evolutionary rate matrices" in table 4. I can understand why they would want to test models of equality, proportionality, and no shared structure between rate matrices. However, I may have missed their explanation of this in the paper but I do not have a grasp of the biological significance of forcing the rate matrices to share eigenvectors. Is this a constraint that one makes sense in biology? I understand why they wanted to show the data but does placing any restrictions on the two matrices make sense if you have already decided that they should be different?