We present optical observations of the recently discovered ROSAT source RX J1238 - 38, which is a new member of the intermediate polar class of asynchronous magnetic cataclysmic variables (CVs). Optical photometry reveals two coherent periodicities at 1860 and 2147 s respectively, with similar amplitudes of similar to 8 per cent. Infrared (J-band) intensity variations are detected only at the 1860-s period, at an amplitude of similar to 15 per cent. The initial hypothesis, that these two periods were the spin and synodic (i.e., beat) period respectively, appears not to be supported by the spectroscopic data. The emission lines vary on the longer photometric period, and radial velocity variations are detected at this period and at a longer period of similar to 5300 s, which we identify as the spin and orbital periods respectively. The most likely explanation for the 1860-s period is that it is the first harmonic of the omega - Omega sideband, leading to an improved determination of the orbital period as 5077 s (= 84 min). If this interpretation is correct, RX J1238 - 38 joins EX Hya as the only other intermediate polar below the 2-3 h period gap, and with an orbital period close to the minimum for CVs with non-degenerate secondaries. The spin-modulated emission-line radial velocities and widths appear to be anticorrelated, with maximum width occurring at maximum blueshift. Such an anticorrelation is expected for aspect changes of accretion curtains. Polarimetric observations of RX J1238 - 38 were inconclusive, although we can put a limit of 0.4 per cent on any variability on the circular polarization, and certainly there is no indication of variations at the photometric or spectroscopic periods.