SW Sextantis stars: the dominant population of CVs with orbital periods between 3-4 hours

[Abridged] We present time-series optical photometry of five new CVs identified by the Hamburg Quasar Survey. The eclipses observed in HS 0129+2933, HS 0220+0603, and HS 0455+8315 provided very accurate orbital periods of 3.35129827(65), 3.58098501(34), and 3.56937674(26) h, respectively. HS 0805+3822 shows grazing eclipses and has a likely orbital period of 3.2169(2) h. Time-resolved optical spectroscopy of the new CVs (with the exception of HS 0805+3822) is also presented. Radial velocity studies provided an orbital period of 3.55 h for HS 1813+6122, which allowed us to identify the observed photometric signal at 3.39 h as a negative superhump wave. The spectroscopic behaviour clearly identifies these new CVs as new SW Sextantis stars. These new additions increase the number of known SW Sex stars to 35. Almost 40 per cent of the total SW Sex population do not show eclipses, invalidating the requirement of eclipses as a defining characteristic of the class and the models based on a high orbital inclination geometry alone. On the other hand, the predominance of orbital periods in the narrow 3-4.5 h range is becoming more pronounced. In fact, almost half the CVs which populate the 3-4.5 h period interval are definite members of the class. These statistics are confirmed by our results from the Hamburg Quasar Survey CVs. Remarkably, 54 per cent of the Hamburg nova-like variables have been identified as SW Sex stars with orbital periods in the 3-4.5 h range. The observation of this pile-up of systems close to the upper boundary of the period gap is difficult to reconcile with the standard theory of CV evolution, as the SW Sex stars are believed to have the highest mass transfer rates among CVs. Finally, we review the full range of common properties that the SW Sex stars exhibit.