Let \(\mathcal H\) be a complex separable Hilbert space. For \(A\in B({\mathcal H})\), denote by \({\mathcal W}(A)\) the smallest algebra containing \(A\) and closed in the weak operator topology. An operator \(A\in B({\mathcal H})\) is called a power partial isometry if \(A^{n}\) is a partial isometry for every positive integer \(n\). The main result of the paper is as follows. Theorem. Let \(A\in B({\mathcal H})\) be a completely non-unitary power partial isometry. Then the following properties are equivalent: (1) \(A\) is reflexive; (2) \(A\) is hyperreflexive; (3) each \(B\in{\mathcal W}(A)\) of rank two generates a one-dimensional ideal.