We present the first time-resolved spectroscopic study of the cataclysmic variable DW Cancri. We have determined an orbital period of 86.10 +/- 0.05 min, which places the system very close to the observed minimum period of hydrogen-rich cataclysmic variables. This invalidates previous speculations of DW Cnc being either a permanent superhumper below the period minimum or a nova-like variable with an orbital period longer than 3 h showing quasi-periodic oscillations. The Balmer and HeI lines have double-peaked profiles and exhibit an intense S-wave component moving with the orbital period. Remarkably, the Balmer and HeI radial velocity curves are modulated at two periods: 86.10 +/- 0.05 (orbital) and 38.58 +/- 0.02 min. The same short period is found in the equivalent width variations of the single-peaked HeII 4686 line. We also present time-resolved photometry of the system which shows a highly coherent variation at 38.51 min, consistent with the short spectroscopic period. The large number of similarities with the short-period intermediate polar V1025 Cen lead us to suggest that DW Cnc is another intermediate polar below the period gap, and we tentatively identify the photometric and spectroscopic 38-min signals with the white dwarf spin period. DW Cnc has never been observed to undergo an outburst, but it occasionally exhibits low states ~2 mag fainter than its typical brightness level of V~= 14.5, resembling the behaviour of the high mass-transfer VY Scl stars.