Abstract <p> We introduce a new doubly unitary operator (isospectry) on wave functions, whose existence is equivalent to the equality of the spectra of two quantum billiards. It generates a map of nested billiards (induced reflection) with rich properties. This allows proving that in addition to isometry there is a unique realization of isospectrality of billiards, namely, a multivalued isometry. Then they are cellular and are constructed by reflections of the same cell. The algebra of their cellular subsets is isomorphic to a complete Boolean algebra and leads to the known generalized double negation formula, which corresponds to the logical foundations of quantum theory. </p>