What is twisted in a twisted Poisson structure? The usual definition of a twisted Poisson algebra involving a 3-form
does not seem to refer to a Poisson algebra that is then twisted,
i.e. twisting either the commutative product or the Lie bracket.
Have I missed where such a discussion occurs?
 A: I'm sure experts will be able to provide more detailed answers.  My reading of the paper http://arxiv.org/pdf/math/0107133v2.pdf introducing twisted Poisson structures indicates that it is the Courant structure on the generalized tangent bundle which is twisted by the 3-form, and that "twisted Poisson structures" are bivectors defining Dirac structures for the twisted Courant structure.
A: The (algebraic) Poisson counterpart of Drinfel'd twist deformation from Hopf algebras is the following: let $u$ be an invertible element of a Poisson algebra $A$. Then $A$, with the same associative multiplication but with the new bracket defined by 
$$
[a, \, b]_u := u^{-1} [ u a, \, u b] 
$$
is a Poisson algebra called the u-conformal deformation of $A$. I deal with this deformation ( http://arxiv.org/abs/1406.3529 ) and seems to be quite intersting. The geometric meanning of it due by C.M. Marle (1991). A very tempting and difficult question that arise in the context is the classification of all u-conformal deformation of a given Poisson algebra. 
