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The discovery of two-dimensional topological photonic systems has transformed our views on the propagation and scattering of electromagnetic waves, and the quest for similar states in three dimensions is open. Here, we theoretically demonstrate that it is possible to design symmetry-protected three-dimensional topological states in an all-dielectric platform, with the electromagnetic duality between electric and magnetic fields being ensured by the structure design. Magneto-electrical coupling plays the role of a synthetic gauge field that determines a topological transition to an ‘insulating’ regime with a complete three-dimensional photonic bandgap. We reveal the emergence of surface states with conical Dirac dispersion and spin-locking, and we numerically confirm robust propagation of the surface states along two-dimensional domain walls with first-principles studies. The proposed system represents a table-top …