We propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric interior penalty discontinuous Galerkin discretization of H(curl,Ω)-elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners rely on the auxiliary space method (ASM) employing an auxiliary space of H(curl,Ω)-conforming finite element functions together with a relaxation technique (local smoothing). On simplicial meshes, the proposed preconditioner enjoys asymptotic optimality with respect to mesh refinement. It is also robust with respect to jumps in the coefficients ν and β in the second- and zeroth-order parts of the operator, respectively, except when the problem changes from curl-dominated to reaction-dominated and vice versa. On quadrilateral/hexahedral meshes some of the proposed ASM solvers may fail, since the related H(curl,Ω)-conforming finite element space does not provide a spectrally accurate discretization. Extensive numerical experiments are included to verify the theory and assess the performance of the preconditioners.