We recall the main results of double shuffle theory: the cyclotomic analogues of MZVs (of order N \geq 1) satisfy a collection of relations arising from the study of their combinatorics, and also from their identifications with periods. The scheme arising from these relations is a torsor Under a prounipotent algebraic group DMR_0. This is a subgroup of the group Out^* of invariant tangential outer automorphisms of a free Lie algebra, equipped with an action of mu_N. The Lie algebra dmr_0 of DMR_0 is a subspace of the Lie algebra out^*, defined by a pair of shuffle relations (Racinet) and containing the Grothendieck-Teichmüller Lie algebra or its analogues (Furusho). We show that the harmonic coproduct may be viewed as an element of a module over out^*, and that dmr_0 then identifies with the stabilizer Lie algebra of this element. A similar identification concerning DMR_0 enables one to construct a "Betti" version of the harmonic coproduct, and to identify the scheme arising from double shuffle relations as the set of elements of Out^* taking the harmonic coproduct to its "Betti" version.