We scrutinize the saddle crossings of a simple cluster of six atoms to show (a) that it is possible to choose a coordinate system in which the transmission coefficient for the classical reaction path is unity at all energies up to a moderately high energy, above which the transition state is chaotic; (b) that at energies just more than sufficient to allow passage across the saddle, all or almost all the degrees of freedom of the system are essentially regular in the region of the transition state; and (c) that the degree of freedom associated with the reaction coordinate remains essentially regular through the region of the transition state, even to moderately high energies. Microcanonical molecular dynamics simulation of Ar-6 bound by pairwise Lennard-Jones potentials reveals the mechanics of passage. We use Lie canonical perturbation theory to construct the nonlinear transformation to a hyperbolic coordinate system which reveals these regularities. This transform "rotates away'' the recrossings and nonregular behavior, especially of the motion along the reaction coordinate, leaving a coordinate and a corresponding dividing surface in phase space which minimize recrossings and mode-mode mixing in the transition state region. The action associated with the reactive mode tends to be an approximate invariant of motion through the saddle crossings throughout a relatively wide range of energy. Only at very low energies just above the saddle could any other approximate invariants of motion be found for the other, nonreactive modes. No such local invariants appeared at energies at which the modes are all chaotic and coupled to one another.