Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group

Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group ${\mathsf{G}\mathsf{R}\mathsf{T}}_1$. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that ${\mathsf{G}\mathsf{R}\mathsf{T}}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.