The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these tools to derive several families of facets. The tools include necessary and sufficient conditions for an inequality to be facet defining, and an algorithm for generating all undominated circuits. We use a novel approach to identifying families of facet-defining inequalities, based on the structure of variable indices rather than on subgraphs such as combs or subtours. This leads to our main result, a hierarchy of families of facet-defining inequalities and polynomial-time separation algorithms for them.