We introduce a novel phase-only diffractive optical element called chiral binary square axicon (CBSA). The CBSA is designed by linearly rotating the square half-period zones of the binary square axicon with respect to one another. A quadratic phase mask (QPM) is combined with the CBSA using modulo-2π phase addition technique to bring the far-field intensity pattern of CBSA at the focal plane of the QPM and to introduce quasiachromatic effects. The periodically rotated zones of CBSA produce a whirlpool phase profile and twisted intensity patterns at the focal plane of QPM. The degree of twisting seen in the intensity patterns is dependent upon the angular step size of rotation of the zones. The intensity pattern was found to rotate around the optical axis along the direction of propagation. The phase patterns of CBSA with different angles of zone rotation are displayed on a phase-only spatial light modulator, and the experimental results were found to match with the simulation results. To evaluate the optical trapping capabilities of CBSA, an optical trapping experiment was carried out and the optical fields generated by CBSA were used for trapping and rotating yeast cells.