  We construct an explicit transitive free action of a Banach space of H\"older
functions on the space of branched rough paths, which yields in particular a
bijection between theses two spaces. This endows the space of branched rough
paths with the structure of a principal homogeneous space over a Banach space
and allows to characterize its automorphisms. The construction is based on the
Baker-Campbell-Hausdorff formula, on a constructive version of the
Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to
describe branched rough paths in terms of anisotropic geometric rough paths.
