Bloch's transformation U:Ω0→Ω from the zeroth-order space for a perturbation problem to the corresponding space of exact eigenvectors, was found as a geometrically defined alternative to the algebraically constructed Van Vleck transformation. Klein's theorem of uniqueness transferred some of this geometrical interpretation to its canonical form Uc=egc. Quite recently Kvaal has taken a large step further by writing Uc as a product of commuting planar rotations, obtained by describing Ω0 and Ω in terms of certain principal vectors and canonical angles. Kvaal's approach is now developed further, using a new commutation relation which simplifies algebraic manipulations substantially. It allows for a simple definition of an operator θ for the angle between Ω0 and Ω which has Kvaal's vectors and angles as eigenvectors and eigenvalues. Klein's theorem is refined in various ways. The impact of the approach on a number of previous results is considered. © 2015 Wiley Periodicals, Inc.