The fact that the mixed second order partial derivatives of a $C^2$ smooth scalar valued function are equal seems, to me, quite surprising. For example, if you interpret $\frac{\partial ^2f}{\partial y \partial x}$ as the change of the slope of a tangent line along the $x$-axis when moving along the $y$-axis, it's not at all obvious to me that this should equal $\frac{\partial ^2f}{\partial x \partial y}$.
Does there exist some intuitive or visual way to explain this equality? Just to be clear, I'm not looking for a formal proof (this can be found in most textbooks), but some intuitive reason or hint as to why this might be true.