We study the isospectrality problem for a relativistic free quantum particle, described by the Dirac Hamiltonian, confined in a one-dimensional ring with a junction. We analyze all the self-adjoint extensions of the Hamiltonian in terms of the boundary conditions at the junction, characterizing the energy spectrum by means of a spectral function. By determining the symmetries of the latter, we are able to divide the self-adjoint extensions in two classes, identifying all the families of isospectral Hamiltonians, and thus completely characterizing the isospectrality problem.