The magnetic response expected from a state characterized by rotating antiferromagnetism in a neutron-scattering experiment is calculated. We predict the occurrence of a peak at the frequency of the rotation of the rotating antiferromagnetic order parameter. The doping dependence of this frequency is very similar to that of the frequency of the magnetic resonance observed in the neutron-scattering experiments for the hole-doped high-$T_C$ cuprates. This leads us to propose the rotating antiferromagnetism as a possible mechanism for this magnetic resonance. We conclude that while the magnitude of the rotating antiferromagnetic order parameter was previously proposed to be responsible for the pseudogap and the unusual thermodynamic and transport properties, the phase of the rotating order parameter is proposed here to be responsible for the unusual magnetic properties of the high-$T_C$ copper-oxide superconductors.