We prove that Poincaré duality algebras are characterized by a certain rigidity property. As a consequence of this fact, we show that the $k$-isomorphism of a Poincaré duality algebra $H^*$ of top dimension $n$ is uniquely determined by the factor $H^*/H^n$, provided that $k$ is algebraically closed. Using this and usual methods of descent theory, we obtain a description of the set of $k$-isomorphism es of Poincaré duality algebras with the same given isomorphism of $H^*/H^n$, for any field $k$ of characteristic zero. These results are then applied to the study of homotopy types of simply connected Poincaré duality spaces.