The DzyaloshinskiiMoriya interaction (DMI) plays a crucial role in the formation of chiral magnetic structures, such as chiral domain walls and magnetic skyrmions. Recent studies have revealed that anisotropic DMI can arise in specific systems or conditions, which is essential for the formation of three-dimensional spin textures. However, the impact of anisotropic DMI on magnetic moment switching has not been comprehensively studied. In this work, we systematically investigate the influence of anisotropic DMI on spin-orbit torque (SOT)-driven magnetization switching, employing a macrospin model to elucidate the underlying mechanisms. Our findings show that anisotropic DMI introduces a pronounced asymmetry in the magnetization reversal process. Simulations based on the LandauLifshitzGilbert equation further demonstrate that anisotropic DMI not only breaks the symmetry of the switching trajectory but also enhances switching efficiency by reducing the switching time. Furthermore, we demonstrate the realization of five distinct logic operations (AND, NAND, OR, NOR, NOT) within a single device, exploiting the asymmetric SOT-driven magnetization switching induced by anisotropic DMI. Overall, our results not only provide a comprehensive understanding of the role of anisotropic DMI in SOT-driven magnetic switching, but also open new avenues for the engineering of next-generation spintronic devices leveraging DMI.