one could establish a basis-independent condition that was equivalent to the statement that a basis exists in which is satisfied. Such a basis-independent condition would then be a necessary condition for a supersymmetric Higgs sector.[^5]

The conditions of do not match any single condition listed in Table \[master1\]. Nevertheless, given a scalar potential whose parameters satisfy , one can *always* transform to a new basis that explicitly satisfies the CP3-symmetry conditions of Table \[master1\], and vice versa. That is, a basis-invariant characterization of a CP3-symmetric THDM would guarantee that some basis exists in which holds. It follows that *the Higgs sector of the MSSM is a CP3-symmetric THDM.*

The proof of the assertions above have been given in Ref. [@FHS]. For completeness, we review the arguments here. We first define the discrete flavor symmetry $\Pi_2$ which corresponds to a THDM scalar potential that is symmetric under the interchange of the two Higgs fields. In this case, the scalar potential parameters satisfy: \[pi2\] m\_[11]{}\^2=m\_[22]{}\^2,m\_[12]{}\^2 ,\_1=\_2,\_5 , \_6=\_7. In Ref. [@FHS], we showed that given a $\Pi_2$-symmetric scalar potential, there exists another basis which is $Z_2$-symmetric. Consequently, we did not list $\Pi_2$ as a separate symmetry in Table \[master1\]. However, if we 