Harnessing instabilities in architected lattices and metamaterials allows for controlled nonlinear deformations, enabling desired mechanical functions like shape morphing and energy absorption. By precisely tailoring instabilities such as buckling into the structure, deformations become a powerful instrument rather than a failure mode, offering new possibilities for predictable responses to mechanical loads. Inspired by the bulk-boundary correspondence in condensed matter physics, an analogous relationship is explored in metamaterials, where the underlying topology dictates whether the structure under load buckles or barrels. The underlying mechanism that originates from a polarization conversion in the elementary beam network is discussed. Moreover, these predictions, which can be extended to more complex topologies and higher dimensions, showcase that a mode inversion process governs both global deformations and the orientation of localized shear strains within the bulk. It is anticipated that the buckle-barrel correspondence can be extended to non-Hookean materials, paving the way for predicting the onset and evolution of true failure.