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We consider the Grothendieck--Teichm\"uller group under a new aspect. Usingreal algebraic geometry and web theory we show that it preserves dihedralsymmetry relations, present in the fundamental groupoids of configurationspaces of marked points on $\mathbb{C}$. The motivation of this paper is to beunderstood in the light of Grothendieck's initial philosophy stating thatthroughout hidden symmetries of the moduli spaces of curves one can shed somelight on the absolute Galois group. This appears as a new development of theconstruction of the avatar of the Grothendieck--Teichm\"uller group andprepares as well the ground for studying further relations to the motivicGalois group.

Hidden symmetries of the Grothendieck--Teichm\"uller group