[en] We analyze various families of discrete maps in N -qubit systems in the context of permutation invariance. We prove that the tomographic condition imposed on the self-dual (Wigner) map is incompatible with the requirement of the invariance under particle permutations (except for the two-qubit case), which makes it impossible to project the Wootters-like Wigner function into the space of symmetric measurements. We also provide several explicit forms of the self-dual mappings: (a) tomographic and (b) permutation invariant, and analyze the symmetric projection in the latter case. (paper)