Geometric intuition behind the Lie bracket of vector fields

I understand the definition of the Lie bracket and I know how to compute it in local coordinates.

But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric intuition ?

For instance, if we take $U = x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$ and $V = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$, should it be obvious that $[U, V] = 0$ ?

One way to get a geometric intuition for the Lie bracket is to note $\Phi_*([U,V])=[\Phi_*(U),\Phi_*(V)]$, i.e. the Lie bracket transforms canonically under the diffeomorphism $\Phi$. Now if we have a straightening $\Phi^U$ of the vector field $U$ (such that $\Phi^U_*(U)$ is constant in our coordinate system), then $[U,V]$ is just the derivative of $V$ along (the constant direction) $U$ in that coordinate system.