Summary: Let \(A\) be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry \(V\) so that \(AV > 0\) and \(A\) is Hankel with respect to \(V\), i.e., \(V^*A = AV\), if and only if \(A\) is not invertible. The isometry \(V\) can be chosen to be isomorphic to \(N \in \mathbb{N} \cup\{+\infty\}\) copies of the unilateral shift if \(A\) has spectral multiplicity at most \(N\). We further show that the set of all isometries \(V\) so that \(A\) is Hankel with respect to \(V\) are in bijection with the set of all closed, symmetric restrictions of \(A^{-1}\).