We report the discovery of a cataclysmic variable that was detected as an X-ray source by the HEAO-1 modulation collimator. The spectrum is typical of a weak-lined nova-like system, with moderately strong He II lambda 4686-Angstrom emission (similar to 0.3H beta) and a very blue continuum. Extensive photometric observations (similar to 94 h in total) from 1989 February to 1990 March reveal the presence of a 3.34-h quasi-sinusoidal, similar to 5 per cent modulation. We confirm with a radial velocity study that this variation occurs at, or very near to, the orbital period. Furthermore, the relative phasing of the photometry and emission-line velocities, and the light curve structure, points to a similar origin to that proposed for several other X-ray-selected CVs (e.g. AO Psc and FO Aqr), namely an X-ray-illuminated disc bright-spot. This and other circumstantial evidence suggest that 1H0551-819 may be an intermediate polar. On suborbital time-scales, all of the light curves are dominated by incoherent flickering, with occasional enhancements of power at periods in the region of 600-2400 s. The short coherence, but with significant amplitude, points to a QPO-like interpretation. The usual optical signature of an intermediate polar, namely a strong and coherent optical variation at the spin or sideband period, is lacking in 1H0551-819. Some spectral and temporal characteristics of 1H0551-819, however, resemble known or strongly suspected intermediate polars (AO Psc, FO Aqr, TV Col, V795 Her, TT Ari), and we suggest that 1H0551-819 may be such a system, but with a suppressed spin-sideband signature.