Let $V$ be an oriented inner product space of dimension $n$. The Hodge star operator maps $\Lambda^k V\to \Lambda^{n-k}V$. In particular it maps $V\to \Lambda^{n-1}V.$ $V$ carries a representation of the rotation group $\text{SO}(n)$ (after choosing a basis, perhaps), and $\Lambda^k V$ carries an induced representation, and we might naïvely expect the Hodge star operator to be a morphism of representations so that $(\Lambda^{n-1}g)\cdot *v=*(g\cdot v).$ However for some reason, an isomorphism with the dual space representation seems more appropriate, which I think should look something like $(\Lambda^{n-1}g)\cdot *v=*((g^{-1})^T\cdot v)$. Is this correct? The Hodge dual is an isomorphism between the standard rep and the dual rep? I only suspect it to be so, because that's what I think it would take to make the Laplacian $\Delta=*d*d$ belong to the trivial representation. But I can't figure out how to show it directly. Is there some nice way to see it?
Edit: The Stackexchange software linked me to a related question with an excellent answer by Qiaochu Yuan. He describes the Hodge star as an isomorphism to the exterior power of the dual space, I hadn't seen it that way before, but it lends credence to the idea that you should get a contragredient representation.