  The M\"obius $(8_4)$ configuration is generalized in a purely combinatorial
approach. We consider $(2n_n)$ configurations ${\goth M}_{(n,\varphi)}$
depending on a permutation $\varphi$ in the symmetric group $S_n$. Classes of
non-isomorphic configurations of this type are determined. The parametric
characterization of ${\goth M}_{(n,\varphi)}$ is given. The uniqueness of the
decomposition of ${\goth M}_{(n,\varphi)}$ into two mutually inscribed
$n$-simplices is discussed. The automorphisms of ${\goth M}_{(n,\varphi)}$ are
characterized for $n\geq 3$.
