If it helps, the following identity holds (in index notation, where repeated indices are summed over): $$[\vec w\times(\vec \nabla\times\vec v)]_\mu = (\partial_\mu v_\nu - \partial_\nu v_\mu)w_\nu. $$ This can be written in matrix notation as $$ \vec w\times(\vec \nabla\times\vec v) = (\mathbf J_\vec v^{\mathsf T} - \mathbf J_\vec v)\vec w $$ where $\mathbf J_\vec v$ is the Jacobian matrix of $\vec v$.

The second term is the negative of the directional derivative of $\vec v$ along $\vec w$. I'm not sure if there is a simple interpretation of the first term, but hopefully someone else can chime in on that.