  We analyze different families of discrete maps\ in the N-qubit systems in the
context of the 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, which makes it impossible to
project the Wootters-like Wigner function into the space of symmetric
measurements. We also provide several \textit{explicit} forms of the self-dual
mappings: a) tomographic and b) permutation invariant \ and analyze the
symmetric projection in the latter case.
