Non-homogeneous Koszul duality is now well-understood. Here are a few references: 

   - I guess the original reference is 

> L. E. Positsel′ski˘ı. Nonhomogeneous
> quadratic duality and curvature.
> Funktsional. Anal. i Prilozhen.,
> 27:57–66, 96, 1993.

   - for a more systematic study you can have alook at 

> A. Polishchuk and L. Positselski.
> Quadratic algebras, volume 37 of
> University Lecture Series. American
> Mathematical Society, Providence, RI,
> 2005.

   - As far as I remember the [new book of Loday and Vallette][1] discusses this too (see $\S 3.6$). 

   - You can find the statement that Weyl and Clifford algebras are Koszul in the inhomogenous sens in [this paper of Braverman-Gaistgory][2] ($\S 5.3$). 

Nevertheless, as it is said in Leonid Positselski's comment, Weyl and Clifford algebra are not Koszul dual to each other. The reason is that inhomogeneous Koszul duality is inhomogeneous!

   - quadratic-linear algebras are dual to DG quadratic algebras (e.g. the universal enelopping algebra of a Lie algebra is Koszul dual its Chevalleay-Eilenberg algebra). 

   - quadratic--linear-constant algebra (e.g. Weyl or Clifford, for which there is even no linear part) are dual to curved quadratic DG algebras. E.g. for the Weyl algebra $\mathcal W_{(V,\omega)}$, its Kozsul dual is the pair $(\wedge(V^*),\omega)$ where the symplectic form $\omega$ is viewed as a curvature (a degree 2 element) in the exterior algebra. 


  [1]: http://math.unice.fr/~brunov/Operads.pdf
  [2]: http://arxiv.org/abs/hep-th/9411113