G. Raby proved in  that the Godbillon-Vey invariant for codimension-one foliations is invariant by -diffeomorphisms. In this paper we generalize the result for foliations of arbitrary codimension.
Taro ASUKE. "Invariance of the Godbillon-Vey class by -diffeomorphisms for higher codimensional foliations." J. Math. Soc. Japan 51 (3) 655 - 660, July, 1999. https://doi.org/10.2969/jmsj/05130655