  We define and study odd and even analogues of the major index statistics for
the classical Weyl groups. More precisely, we show that the generating
functions of these statistics, twisted by the one-dimensional characters of the
corresponding groups, always factor in an explicit way. In particular, we
obtain odd and even analogues of Carlitz's identity, of the Gessel-Simion
Theorem, and a parabolic extension, and refinement, of a result of Wachs.
