We present a detailed analysis of the kinematics of 112 stars that mostly comprise the high-velocity S cluster and orbit the supermassive black hole Sgr A* at the center of the Milky Way. For 39 of them, orbital elements are known; for the remainder, we know proper motions. The distribution of the inclinations and the proper motion flight directions deviate significantly from a uniform distribution, which one expects if the orientation of the orbits are random. Across the central arcseconds, the S-cluster stars are arranged in two almost edge-on disks that are located at a position angle approximately +/-45{deg} with respect to the Galactic plane. The angular momentum vectors for stars in each disk point in both directions, i.e., the stars in a given disk rotate in opposite ways. The poles of this structure are located only about 25{deg} from the line of sight. This structure may be the result of a resonance process that started with the formation of the young B-dwarf stars in the cluster about 6Myr ago. Alternatively, it indicated the presence of a disturber at a distance from the center comparable to the distance of the compact stellar association IRS 13.