Adjoint of the Hodge star on middle-degree forms (${\ast }^{†}=(-1{)}^{m^2}\ast$)

On $m$-forms in an oriented Riemannian manifold of dimension $2m$, the Hodge star operator is orthogonal and satisfies ${\ast }^2=(-1{)}^{m^2}$. HenceIt is self-adjoint when $m$ is even and skew-adjoint when $m$ is odd.