  We define and study odd analogues of classical geometric and combinatorial
objects associated to permutations, namely odd Schubert varieties, odd
diagrams, and odd inversion sets. We show that there is a bijection between odd
inversion sets of permutations and acyclic orientations of the Tur\'{a}n graph,
that the dimension of the odd Schubert variety associated to a permutation is
the odd length of the permutation, and give several necessary conditions for a
subset of $[n] \times [n]$ to be the odd diagram of a permutation. We also
study the sign-twisted generating function of the odd length over descent
classes of the symmetric groups.
