We report on results concerning the polyhedral structure of, and integer linear programming formulations for, the quadratic linear ordering problem. Specifically, we provide a deeper analysis of the characteristic equation system that takes part in the minimal description of the convex hull of its feasible solutions, and we determine an accessible description of a restricted and inextensible subset of the odd-cycle inequalities that induces facets of it. We also present an extended formulation in which the products of the linear ordering variables that share an index are linearized implicitly.