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Let $A$ be a bounded, injective and self-adjoint linear operator on a complexseparable 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 $A$ hasspectral multiplicity at most $N$. We further show that the set of allisometries, $V$, so that $A$ is Hankel with respect to $V$, are in bijectionwith the set of all closed, symmetric restrictions of $A^{-1}$.

On unitary equivalence to a self-adjoint or doubly-positive Hankel operator