  Principal angles are used to define an angle bivector of subspaces, which
fully describes their relative inclination. Its exponential is related to the
Clifford geometric product of blades, gives rotors connecting subspaces via
minimal geodesics in Grassmannians, and decomposes giving Pl\"ucker
coordinates, projection factors and angles with various subspaces. This leads
to new geometric interpretations for this product and its properties, and to
formulas relating other blade products (scalar, inner, outer, etc., including
those of Grassmann algebra) to angles between subspaces. Contractions are
linked to an asymmetric angle, while commutators and anticommutators involve
hyperbolic functions of the angle bivector, shedding new light on their
properties.
