  The scalar potential of the N-Higgs-doublet model (NHDM) is best analyzed not
in the space of N complex doublets $\phi_a$ but in the $N^2$-dimensional space
of real-valued bilinears constructed of $\phi_a^\dagger \phi_b$. In particular,
many insights have been gained into CP violation in the 2HDM and 3HDM by
studying how generalized CP transformations (GCPs) act in this bilinear space.
These insights relied on the fact that GCPs, which involved an odd number of
mirror reflection, could be clearly distinguished from Higgs family
transformations by the sign of the determinant of the transformation matrix. It
was recently pointed out that this criterion fails starting from 4HDM, where
the reflection/rotation dichotomy does not exist anymore. In this paper, we
restore intuition by finding a different quantity which faithfully
discriminates between GCPs and Higgs family transformations in the bilinear
space for any number of Higgs doublets. We also establish the necessary and
sufficient conditions for an orthogonal transformation in the bilinear space to
represent a viable transformation back in the space of N doublets, which is
helpful if one prefers to build an NHDM directly in the bilinear space.
