In this work, we consider the geomagnetic field inversion model proposed by Gissinger et al. [Europhys. Lett. 90(4), 49001 (2010)], where a quadratic term is added for symmetry control purposes. The resulting system is explored in both symmetric and asymmetric modes of operation. In the symmetric case, we report a bursting phenomenon and heterogeneous multistability of six and four different attractors. We show that the model owns an offset adjustment feature. In the asymmetric case, the model develops different phenomena, such as the coexistence of (four and three) asymmetric attractors, asymmetric (periodic and chaotic) bursting oscillation, and transient asymmetric bursting phenomenon. The effect of symmetry breaking is also manifested in the bubbles of bifurcation. It is shown that this system can leave from the multistable state to a monostable state by adjusting the coupling parameter of a linear controller. Moreover, microcontroller-based implementation of the system is considered to check the correctness of the numerical results.