We measure the 3D kinematic structures of the young stars within the central 0.5pc of our Galactic Center using the 10m telescopes of the W. M. Keck Observatory over a time span of 25yr. Using high-precision measurements of positions on the sky and proper motions and radial velocities from new observations and the literature, we constrain the orbital parameters for each young star. Our results show two statistically significant substructures: a clockwise stellar disk with 18 candidate stars, as has been proposed before, but with an improved disk membership; and a second, almost edge-on plane of 10 candidate stars oriented east-west on the sky that includes at least one IRS 13 star. We estimate the eccentricity distribution of each substructure and find that the clockwise disk has <e>=0.39 and the edge-on plane has <e>=0.68. We also perform simulations of each disk/plane with incompleteness and spatially variable extinction to search for asymmetry. Our results show that the clockwise stellar disk is consistent with a uniform azimuthal distribution within the disk. The edge-on plane has an asymmetry that cannot be explained by variable extinction or incompleteness in the field. The orientation, asymmetric stellar distribution, and high eccentricity of the edge-on plane members suggest that this structure may be a stream associated with the IRS 13 group. The complex dynamical structure of the young nuclear cluster indicates that the star formation process involved complex gas structures and dynamics and is inconsistent with a single massive gaseous disk.