When is a polynomial in three variables cylindrical?

We prove that a nonconstant polynomial $f\in \mathbb{C}[x_1,x_2,x_3]$ is cylindrical (that is, $f\in \mathbb{C}[x_1,x_2]$ after an invertible linear change of coordinates) if and only if its bordered Hessian determinant vanishes identically. The same conclusion holds over $\mathbb{R}$. We also give explicit counterexamples to this criterion in every dimension $n\ge 4$.