Building on a result by W. Rump, we show how to exploit the right-cyclic law
(x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and
monoids attached with (involutive nondegenerate) set-theoretic solutions of the
Yang-Baxter equation. We develop a sort of right-cyclic calculus, and use it to
obtain short proofs for the existence both of the Garside structure and of the
I-structure of such groups. We describe finite quotients that exactly play for
the considered groups the role that Coxeter groups play for Artin-Tits groups
