We make a systematic analysis of the XMM-Newton X-ray spectra of intermediate polars (IPs) and find that, contrary to the traditional picture, most show a soft blackbody component. We compare the results with those from AM Her stars and deduce that the blackbody emission arises from reprocessing of hard X-rays, rather than from the blobby accretion sometimes seen in AM Hers. Whether an IP shows a blackbody component appears to depend primarily on geometric factors: a blackbody is not seen in those that have accretion footprints that are always obscured by accretion curtains or are only visible when foreshortened on the white-dwarf limb. Thus we argue against previous suggestions that the blackbody emission characterises a separate sub-group of IPs which are more akin to AM Hers, and develop a unified picture of the blackbody emission in these stars.