Quadratic Form with perpendicular vectors

Question

I have a function $f:\mathbb{R}^n\to\mathbb{R}$ with normalized gradient $\widehat{\nabla f(x)}$ and Hessian $H(x)$ at the point $x\in\mathbb{R}^n$. I also know that $\widehat{y}$ is a unit vector perpendicular to the unit gradient $\widehat{y}\perp\widehat{\nabla f(x)}$. Can I say anything interesting about the following quadratic form? $$ \widehat{\nabla f(x)}^\top H(x) \widehat{y} $$ There is so much structure in this problem that there must be something special about this, perhaps a bound (assuming more things) or a nice expression.