  This paper is the first in a series which aims at: (a) giving a proof that
the associator relations between multizeta values imply the double shuffle and
regularization (DSR) ones, alternative to that of the second-named author's
2010 paper; (b) enhancing Racinet's construction of a torsor structure over the
Q-scheme of DSR relations to an explicit bitorsor structure. In this paper, we
revisit Racinet's original DSR formalism, whose main character is an algebra
coproduct, called the harmonic coproduct, and we introduce a variant which is a
module coproduct; we explain the `de Rham' nature of this formalism and
construct a `Betti' counterpart of it; we show how both formalisms can be
interpreted in terms of geometry, following the ideas of Deligne and Terasoma's
unfinished 2005 preprint; we use Bar-Natan's interpretation of associators as
functors from the category of parenthesized braids to that of chord diagrams to
show that any associator relates the Betti and de Rham geometric objects, both
in the `algebraic' and in the `module' setups; we derive that any associator
relates the Betti and de Rham algebra coproducts, as well as their module
counterparts. These results will be used in the next parts of the series.
