A high-level reason that $u \cdot (v \times w) = (u \times v) \cdot w$? I can do the algebra to show that for $u, v, w \in \mathbb{R}^3$, this identity is true:
$$u \cdot (v \times w) = (u \times v) \cdot w$$
But is there a more high-level reason? I didn't expect the cross and dot product to be connected in this surprising way.

Yes. The volume of the parallelepipid is equal to the scalar triple product. Consider the geometric definitions of the cross-product and inner-product to see this.
