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The magnetization switching dynamics of biaxial single-domain homogeneous ferromagnetic particles, in which the two easy axes are perpendicular to each other, is modeled by a four-state clock model and studied by large-scale dynamic Monte Carlo simulations and analytic theory. A zero-field mapping of the statics between the symmetric four-state clock model and two decoupled Ising models is extended to nonzero-field statics and to the dynamics. This significantly simplifies the analysis of the simulation results. We measure the magnetization switching time of the model and analyze the results by droplet theory. The switching dynamics in the asymmetric model is more complicated. If the easy axis is perpendicular to the stable magnetization direction, the system can switch its magnetization via two different channels, one very fast and the other very slow. A maximum value for the switching field as a function of system size is obtained. The asymmetry affects the switching fields differently, depending on whether the switching involves one single droplet or many droplets of spins in the stable magnetization configuration. The angular dependence of the switching field in symmetric and asymmetric models is also studied.