For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $\operatorname{GL}(n/2,\mathbb{C})$ and $\operatorname{Sp}(n)$ structure on $M$, respectively. With additional integrability (the vanishing of the Nijenhuis tensor or the closedness of the almost symplectic form) one has a complex or symplectic structure on $M$.

Since CR and Contact geometry are the odd-dimensional cousins of the above geometries, it seems as though it would make sense for CR and Contact structures to come from a reduction of the structure group of the manifold. However, these structures tend to be defined more in terms of their integrability conditions. I suppose what I am really hoping for is a more unified theory among the different geometric structures. The Riemannian, Almost Symplectic, and Almost Complex structures share an intimacy with their "2 out of 3" property arising from properties of the intersections of the reduced structure groups, and it would be nice if the relationship between Almost Contact and Almost CR structures stemmed from a similar place, and furthermore that their integrability conditions corresponded to some sort of "flatness" on the $G$-structure.

So I ask, do these $G$-structure correspondences exist for the above geometries, and do they share an analogous relationship with their even-dimensional counterparts?