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This paper is the first in a series which aims at: (a) giving a proof thatthe associator relations between multizeta values imply the double shuffle andregularization (DSR) ones, alternative to that of the second-named author's2010 paper; (b) enhancing Racinet's construction of a torsor structure over theQ-scheme of DSR relations to an explicit bitorsor structure. In this paper, werevisit Racinet's original DSR formalism, whose main character is an algebracoproduct, called the harmonic coproduct, and we introduce a variant which is amodule coproduct; we explain the `de Rham' nature of this formalism andconstruct a `Betti' counterpart of it; we show how both formalisms can beinterpreted in terms of geometry, following the ideas of Deligne and Terasoma'sunfinished 2005 preprint; we use Bar-Natan's interpretation of associators asfunctors from the category of parenthesized braids to that of chord diagrams toshow that any associator relates the Betti and de Rham geometric objects, bothin the `algebraic' and in the `module' setups; we derive that any associatorrelates the Betti and de Rham algebra coproducts, as well as their modulecounterparts. These results will be used in the next parts of the series.

The Betti side of the double shuffle theory. I. The harmonic coproducts