I'm having trouble understanding the relationship between the Hessian and the directional derivative.
If given $\bf H$ as the Hessian of some function $f(\bf x)$, and $u$ some unit direction vector, from this answer, the second derivative of $f$ in the direction of $u$ is $\partial_{uu}^2f({\bf x})=u^T{\bf H}u$. I understand that this is essentially the rate of change of the directional derivative $\partial_u f({\bf x})=\nabla f(x) \cdot u $ in the direction of $u$.
$u^T{\bf H}u$ appears say that $\partial_{uu}^2f({\bf x})$ does not change if $u$ is replaced with $-u$. Why is this the case intuitively? I had thought that if the directional derivative increases as we move in the direction of $u$, that the directional derivative would decrease as we move in the opposite direction?