The SW Sex stars are a class of cataclysmic variables, originally identified because they shared a number of enigmatic properties - most notably, single-peaked emission lines instead of the double-peaked lines one would expect from their high-inclination accretion discs. We present high time-resolution spectrophotometry of the eclipsing nova-like variables SW Sex and DW UMa, two of the founding members of the SW Sex class. Both systems show single-peaked Balmer and HeII 4686A emission lines that appear to originate from a region in the disc that lies close to, but downstream of, the bright spot. The emission-line light curves are consistent with the finding from X-ray and ultraviolet observations that we predominantly see the flared disc rim and the unobscured back portion of the disc in these systems. In DW UMa, the HeII 4686A emission line originates from close to the white dwarf and exhibits flaring. Such flares have been used to argue for magnetically-channelled accretion, as in the intermediate polars, but the lack of a clear periodicity in the flares argues for a simpler model in which we are viewing the central regions of the disc through the non-uniform upper edge of a flared disc rim. We also observe narrow, blue-shifted, transient absorption features in the Balmer lines of DW UMa, which we attribute to blobs of material ejected from the system, possibly by a magnetic propeller, that happen to be passing between us and the binary. Our results suggest that the solution to the SW Sex enigma is a combination of dominant bright-spot emission and a self-occulting disc. We also propose a simplified classification scheme for nova-like variables.