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To describe the double shuffle relations between multiple polylogarithmvalues at $N$th roots of unity, Racinet attached to each finite cyclic group$G$ of order $N$ and each group embedding $\iota : G \to \mathbb{C}^{\times}$,a $\mathbb{Q}$-scheme $\mathsf{DMR}^{\iota}$ which associates to eachcommutative $\mathbb{Q}$-algebra $\mathbf{k}$, a set$\mathsf{DMR}^{\iota}(\mathbf{k})$ that can be decomposed as a disjoint unionof sets $\mathsf{DMR}^{\iota}_{\lambda}(\mathbf{k})$ with $\lambda \in\mathbf{k}$. He also exhibited a $\mathbb{Q}$-group scheme $\mathsf{DMR}_0^G$and showed that $\mathsf{DMR}^{\iota}_{\lambda}(\mathbf{k})$ is a torsor forthe action of $\mathsf{DMR}_0^G(\mathbf{k})$. Then, Enriquez and Furusho showedfor $N=1$ that a subscheme $\mathsf{DMR}^{\iota}_{\times}$ of$\mathsf{DMR}^{\iota}$ is a torsor of isomorphisms relating de Rham and Bettiobjects. In previous work, we reformulated Racinet's construction in terms ofcrossed products and identified his coproduct with a coproduct$\widehat{\Delta}^{\mathcal{M}, \mathrm{DR}}_G$ defined on a module$\widehat{\mathcal{M}}_G^{\mathrm{DR}}$ over an algebra$\widehat{\mathcal{W}}_G^{\mathrm{DR}}$ equipped with its own coproduct$\widehat{\Delta}^{\mathcal{W}, \mathrm{DR}}_G$. In this paper, we provide ageneralization of Enriquez and Furusho's result to any $N \geq 1$: we exhibit amodule $\widehat{\mathcal{M}}_N^{\mathrm{B}}$ over an algebra$\widehat{\mathcal{W}}_N^{\mathrm{B}}$ and show the existence of compatiblecoproducts $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$ and$\widehat{\Delta}^{\mathcal{M}, \mathrm{B}}_N$ such that$\mathsf{DMR}^{\iota}_{\times}$ is contained in the torsor of isomorphismsrelating $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$ (resp.$\widehat{\Delta}^{\mathcal{M}, \mathrm{B}}_N$) to$\widehat{\Delta}^{\mathcal{W}, \mathrm{DR}}_G$ (resp.$\widehat{\Delta}^{\mathcal{M}, \mathrm{DR}}_G$).

The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts