The total angular momentum current can be decomposed into orbital and spin contributions in different ways, known as pseudo-gauges. This freedom leads to ambiguities in the definition of local-equilibrium density operators, which in turn affect estimates of spin polarization in heavy-ion collisions. In this work, the pseudo-gauge ambiguity, together with other ambiguities associated with improvements of conserved currents, is reformulated in terms of spurious symmetries corresponding to conserved currents with vanishing total charge. A prescription for the unambiguous definition of a local-equilibrium density operator, defined as deviating from global equilibrium only by small gradients, is introduced using the currents associated with genuine symmetries. The resulting density operator is invariant under transformations that add improvement terms to local currents, including the energy-momentum tensor.