Divergence of cross product of two vectors, intuition

Question

What is the geometrical/physical intuition behind this identity? (possibly in relation to the intuitions commonly given for curl, divergence and cross product) $$\nabla\cdot\left ( \vec{A}\times\vec{B} \right )=\vec{B}\cdot\left ( \nabla \times\vec{A} \right )-\vec{A}\cdot\left ( \nabla \times\vec{B} \right )$$