Spectroscopic search for new SW Sextantis stars in the 3-4 hour orbital period range -- I

[Abridged] We report on time-resolved optical spectroscopy of ten non-eclipsing nova-like cataclysmic variables in the orbital period range between 3 and 4 hours. Of the ten systems so far observed, HL Aqr, BO Cet, AH Men, V380 Oph, AH Pic, and LN UMa are identified as new members of the SW Sex class. We present improved orbital period measurements for HL Aqr (Porb = 3.254 +- 0.001 h) and V380 Oph (Porb = 3.69857 +- 0.00002 h). BO Cet and V380 Oph exhibit emission-line flaring with periodicities of 20 min and 47 min, respectively. The Halpha line of HL Aqr shows significant blueshifted absorption modulated at the orbital period. Similarly to the emission S-wave of the high-inclination SW Sex stars, this absorption S-wave has its maximum blue velocity at orbital phase ~0.5. We estimate an orbital inclination for HL Aqr in the range 19 < i < 27 deg, which is much lower than that of the emission-dominated, non-eclipsing SW Sex stars (i ~ 60-70 deg). This gives rise to the interesting possibility of many low-inclination nova-likes actually being SW Sex stars, but with a very different spectroscopic appearance as they show significant absorption. The increasing blueshifted absorption with decreasing inclination points to the existence of a mass outflow with significant vertical motion. This six new additions to the SW Sex class increase the presence of non-eclipsing systems to about one third of the whole SW Sex population, which therefore makes the requirement of eclipses as a defining criterion for SW Sex membership no longer valid. The statistics of the cataclysmic variable population in the vicinity of the upper period gap is also discussed.