We present complementary data on five intermediate-redshift (0:44 z 0:66) Mg ii-absorbing galaxies, combining high spatial resolution imaging from the Hubble Space Telescope, high-resolution quasi-stellar object (QSO) spectroscopy from the Keck High Resolution Echelle Spectrometer, and galaxy kinematics from intermediate-resolution spectroscopy using the Keck Low-Resolution Imaging Spectrometer. These data allow a direct comparison of the kinematics of gas at large galactocentric impact parameters with the galaxy kinematics obtained from the faint galaxy spectroscopy. All five galaxies appear to be relatively normal spirals, with measured rotation curves yielding circular velocities in the range 100 v c 260 km s À1 . The QSO sight lines have projected impact parameters to the absorbing galaxies in the range 14:5 h À1 kpc d 75 h À1 kpc; the galaxies have inclination angles with respect to the line of sight ranging from 40 to 75 . We find that in four of the five cases examined, the velocities of all of the Mg ii-absorption components lie entirely to one side of the galaxy systemic redshift. The fifth case, for which the galaxy is much less luminous than the other four, has narrow absorption centered at zero velocity with respect to systemic, despite having the largest disk inclination angle in the sample. These observations are consistent with rotation being dominant for the absorbing-gas kinematics; however, the total range of velocities observed is inconsistent with simple disk rotation in every case. Simple kinematic models that simultaneously explain both the systemic offset of the absorbing material relative to the galaxy redshifts and the total velocity width spanned by the absorption require either extremely thick rotating gas layers, rotation velocities that vary with z-height above the extrapolation of the galactic plane, or both. In any case, our small sample suggests that rotating '' halo '' gas is a common feature of intermediate-redshift spiral galaxies and that the kinematic signature of rotation dominates over radial infall or outflow even for gas well away from the galactic plane. We discuss possible explanations for this behavior and compare our observations with possible local analogs.