This paper explores the structure groups \(G_{(X,r)}\) of finite non-degenerate set-theoretic solutions \((X,r)\) to the Yang-Baxter equation. Namely, we construct a finite quotient \(\overline{G}_{(X,r)}\) of \(G_{(X,r)}\), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if \(X\) injects into \(G_{(X,r)}\), then it also injects into \(\overline{G}_{(X,r)}\). We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of \(G_{(X,r)}\). We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free.