The elucidation of the pseudogap phenomenon of the high-transition-temperature (high-T(c)) copper oxides-a set of anomalous physical properties below the characteristic temperature T* and above T(c)-has been a major challenge in condensed matter physics for the past two decades. Following initial indications of broken time-reversal symmetry in photoemission experiments, recent polarized neutron diffraction work demonstrated the universal existence of an unusual magnetic order below T* (refs 3, 4). These findings have the profound implication that the pseudogap regime constitutes a genuine new phase of matter rather than a mere crossover phenomenon. They are furthermore consistent with a particular type of order involving circulating orbital currents, and with the notion that the phase diagram is controlled by a quantum critical point. Here we report inelastic neutron scattering results for HgBa(2)CuO(4+δ) that reveal a fundamental collective magnetic mode associated with the unusual order, and which further support this picture. The mode's intensity rises below the same temperature T* and its dispersion is weak, as expected for an Ising-like order parameter. Its energy of 52-56 meV renders it a new candidate for the hitherto unexplained ubiquitous electron-boson coupling features observed in spectroscopic studies.