The authors present results of an X-ray scattering study on the rotator phases of normal alkanes: CH₃-(CH₂)_n-2-CH₃ (20<or=n<or=33). These are layered plastic-crystal phases in which the long axes of the molecules are all parallel and can be either parallel to or tilted from the layer normal. They have characterized a new tilted rotator phase and measured the distortion, area per molecule and tilt parameters in all five rotator phases. There is no strong even-odd chain effect within the rotator phases, but there is a significant change in phase sequence in the C₂₆-C₂₇ carbon number region. A typical phase sequence for the shorter chains is L-R_II-R_I-R_V-crystal. For longer chains the sequence is L-R_IV-R_III-crystal. In the R_V, R_IV and R_III phases the molecules are tilted. There is a general trend towards decreasing distortion and increasing magnitude of tilt with increasing carbon number. An unusual decrease in area per molecule with increasing temperature for the high-carbon-number phases suggests that defects may play a role in the R_IV phases.