Elucidating the nature of the magnetic ground state of iron-based superconductors is of paramount importance in unveiling the mechanism behind their high-temperature superconductivity. Until recently, it was thought that superconductivity emerges only from an orthorhombic antiferromagnetic stripe phase, which can in principle be described in terms of either localized or itinerant spins. However, we recently reported that tetragonal symmetry is restored inside the magnetically ordered state of certain hole-doped compounds, revealing the existence of a new magnetic phase at compositions close to the onset of superconductivity. Here, we present Mössbauer data that show that half of the iron sites in this tetragonal phase are non-magnetic, establishing conclusively the existence of a novel magnetic ground state with a non-uniform magnetization that is inconsistent with localized spins. Instead, this state is naturally explained as the interference between two commensurate spin-density waves, a rare example of collinear double-Q magnetic order. Our results demonstrate the itinerant character of the magnetism of the iron pnictides, and the primary role played by magnetic degrees of freedom in determining their phase diagram.