We present a kinematic analysis of the globular cluster (GC) system of the giant elliptical galaxy NGC 4365 and find several distinct kinematic substructures. This analysis is carried out using radial velocities for 269 GCs, obtained with the DEIMOS (DEep Imaging Multi-Object Spectrograph) instrument on the Keck II telescope as part of the SAGES Legacy Unifying Globulars and Galaxies Survey (SLUGGS). We find that each of the three (formerly identified) GC colour subpopulations reveal distinct rotation properties. The rotation of the green GC subpopulation is consistent with the bulk of NGC 4365's stellar light, which 'rolls' about the photometric major axis. The blue and red GC subpopulations show 'normal' rotation about the minor axis. We also find that the red GC subpopulation is rotationally dominated beyond 2.5 arcmin (∼17 kpc) and that the root mean squared velocity of the green subpopulation declines sharply with radius suggesting a possible bias towards radial orbits relative to the other GC subpopulations. Additionally, we find a population of low-velocity GCs that form a linear structure running from the SW to the NE across NGC 4365 which aligns with the recently reported stellar stream towards NGC 4342. These low-velocity GCs have g' - i' colours consistent with the overall NGC 4365 GC system but have velocities consistent with the systemic velocity of NGC 4342. We discuss the possible formation scenarios for the three GC subpopulations as well as the possible origin of the low-velocity GC population.