We present time-series optical photometry of five new cataclysmic variables (CVs) identified by the Hamburg Quasar Survey (HQS). The deep eclipses observed in HS 0129+2933 (= TT Tri), 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 of the Balmer emission lines 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 exhibited by all the systems clearly identifies them as new SW Sextantis (SW Sex) stars. HS 0220+0603 shows unusual NII and SiII emission lines suggesting that the donor star may have experienced nuclear evolution via the CNO cycle. These five new additions to the class 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, as more SW Sex stars are identified, 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. The dominance of SW Sex stars is even stronger in the 2-3 h period gap, where they make up 55 per cent of all known gap CVs. These statistics are confirmed by our results from the HQS 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. Only a comprehensive study of this rich phenomenology will prompt to a full understanding of the phenomenon and its impact on the evolution of CVs and the accretion processes in compact binaries in general.