  We study a new kind of Courant algebroid on Poisson manifolds, which is a
variant of the generalized tangent bundle in the sense that the roles of
tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect
product of $\beta$-diffeomorphisms and $\beta$-transformations. It is a
starting point of an alternative version of the generalized geometry based on
the cotangent bundle, such as Dirac structures and generalized Riemannian
structures. In particular, $R$-fluxes are formulated as a twisting of this
Courant algebroid by a local $\beta$-transformations, in the same way as
$H$-fluxes are the twist of the generalized tangent bundle. It is a $3$-vector
classified by Poisson $3$-cohomology and it appears in a twisted bracket and in
an exact sequence.
