There exists increasing evidence that the phase diagram of the high-transition temperature (T c) cuprate superconductors is controlled by a quantum critical point. According to one distinct theoretical proposal, on decreasing the hole-carrier concentration a transition occurs to an ordered state with two circulating orbital currents per CuO 2 square. Below the 'pseudogap' temperature T* (T* >T c), the theory predicts a discrete order parameter and two weakly-dispersive magnetic excitations in structurally simple compounds which should be measurable by neutron scattering. Indeed, novel magnetic order and one such excitation were recently observed. Here, we demonstrate for tetragonal HgBa 2 CuO 4+δ the existence of a second excitation with local character, consistent with the theory. The excitations mix with conventional antiferromagnetic fluctuations, which points towards a unifying picture of magnetism in the cuprates that will probably require a multi-band description.