In this study, we introduce what we believe is a novel holographic optical element called a chiral square Fresnel zone plate (CSFZP). The chirality is imposed on a square Fresnel zone plate (SFZP) using a nonclassical technique by rotating the half-period zones relative to one another. The rotation of the half-period zones, in turn, twists the side lobes of the diffraction pattern without altering the focusing properties inherent to a SFZP. As a consequence, the beam profile is hybrid, consisting of a strong central Gaussian focal spot with gradient force similar to that generated by a lens and twisted side lobes with orbital angular momentum. The optical fields at the focal plane were calculated and found to possess a whirlpool-phase profile and a twisted intensity profile. Analysis of the field variation along the direction of propagation revealed a spiraling phase and amplitude distribution. Poynting vector plot of the fields revealed the presence of angular momentum in the regions of chiral side lobes. The phase of the CSFZPs were displayed on a phase-only reflective spatial light modulator and illuminated using a laser. The intensity patterns recorded in the experiment match the calculated ones, with a strong central focal spot and twisted side lobes. The beam pattern was implemented in an optical trapping experiment and was found to possess particle trapping capabilities.