When discussing Hodge numbers, the identity $h^{p,q} = h^{n-p,n-q}$ is called Serre duality. AFAIU, this identification follows from applying the Hodge $*$ isomorphism. In particular, $H^q(X,\Omega^p_X)$ is dual to $H^{n-q}(X,\Omega^{n-p}_X)$.
If $X$ is now projective, we know that $H^q(X,\Omega_X^p)$ is dual to $H^{n-q}(X, \Omega_X^n \otimes \wedge^p T_X )$. I would like to then deduce the identity of Hodge numbers above by saying that $\Omega_X^{n-p}$ is isomorphic to $\Omega_X^n \otimes \wedge^p T_X$ but I don't know why that should be true.
It would certainly follow if I knew that the wedging map $\Omega^p_X \otimes \Omega^{n-p}_X \to \Omega^n_X$ were an isomorphism.