The dependence of the form of different mathematical depictions of fourfold magnetic anisotropies has been examined, using a simple macro-spin model. Strong differences in longitudinal and transverse hysteresis loops occur due to deviations from the usual phenomenological model, such as using absolute value functions. The proposed possible models can help understanding measurements on sophisticated magnetic nanosystems, like exchange bias layered structures employed in magnetic hard disk heads or magnetic nano-particles, and support the development of solutions with specific magnetization reversal behavior needed in novel magneto-electronic devices.