The paper is concerned with the three-dimensional magnetohydrodynamic equations in the rotational framework concerning with fluid flow of Earth's core and the variation of the Earth's magnetic field. By establishing new balances between the regularizing effects arising from viscosity dissipation and magnetic diffusion with the dispersive effects caused by the rotation of the Earth, we obtain the global existence and uniqueness of solutions of the Cauchy problem of the three-dimensional rotating magnetohydrodynamic equations in Besov spaces. Moreover, the spatial analyticity of solutions is verified by means of the Gevrey class approach.