Suppose $M$ is an arbitrary smooth manifold and $D$ is its bundle of $1$-densities. On the category of finite-dimensional vector bundles over $M$ and linear differential operators between them. There is a contravariant endofunctor that sends a vector bundle $E$ to $E*⊗D$ and a differential operator $f: E→F$ to the adjoint differential operator $f*: F*⊗D→E*⊗D$.

Applying this endofunctor to the standard de Rham (cochain) complex $$0→Ω^0(M)→Ω^1(M)→⋯→Ω^n(M)→0$$ with morphisms being de Rham differentials we obtain another (chain) complex $$0←Λ^0(M)⊗D←Λ^1(M)⊗D←⋯←Λ^n(M)⊗D←0$$ with morphisms being codifferentials. Here $Λ^k(M)$ denotes the bundle of $k$-polyvectors ($k$th exterior power of the tangent bundle).

What is the exact relationship between the homology of this complex and the usual singular (co)homology of $M$?

Using Hodge duality we can rewrite this complex as $$0←Ω^n(M)⊗W←Ω^{n-1}(M)⊗W←⋯←Ω^0(M)⊗W←0$$ where $W$ is the orientation bundle.

It looks like the answer should be some standard fact from the 1950s, therefore any references will be appreciated.