A new light gauge boson U may have both vector and axial couplings. In a large class of theories, however, the new U(1) current JμF naturally combines with the weak neutral current JμZsm, both parity-violating, into a vectorial current JμU, combination of the B,L and electromagnetic currents with a possible dark-matter current. Uμ may be expressed equivalently as cosξCμ+sinξZμsm (“mixing with the Z”) or (1/cosχ)C^μ+tanχAμ (“mixing with the photon”), with C^ coupled to B,L and dark matter. The U boson may be viewed as a generalized dark photon, coupled to SM particles through QU=Q+λBB+λiLi, with strength g′′cosξcos2θ=etanχ . “Kinetic-mixing” terms, gauge invariant or not, simply correspond to a description in a non-orthogonal field basis (rather than to a new physical effect), with the dark photon in general also coupled to B and L. In a grand-unified theory QgutU=Q−2(B−L) at the GUT scale for SM particles, invariant under the SU(4) electrostrong symmetry group, with a non-vanishing ϵ=tanχ already present at the GUT scale, leading to QU≃Q−1.64(B−L) at low energy. This also applies, for a very light or massless U boson, to a new long-range force, which could show up through apparent violations of the equivalence principle.