We offer a geometric interpretation of attractor theories with singular kinetic terms as a union of multiple canonical models. We demonstrate that different domains (separated by poles) can drastically differ in their phenomenology. We illustrate this with the help of a "master model'' that leads to distinct predictions depending on which side of the pole the field evolves before examining the more realistic example of α-attractor models. Such models lead to quintessential inflation within the poles when featuring an exponential potential. However, beyond the poles, we discover a novel behaviour: the scalar field responsible for the early-time acceleration of the Universe reaches the boundary of the field-space manifold, indicating that the theory is incomplete and that a boundary condition must be imposed in order to determine the late-time behaviour of such theories. Turning to multifield models with singular kinetic terms, we see that poles generalise straightforwardly to singular curves, which act as "model walls'' between distinct pole-free inflationary models. As an example, we study a simple two-field α-attractor-inspired model, whose evolution of isocurvature perturbations is sensitive to where the non-canonical field begins its trajectory. We finally discuss initial conditions in attractor theories, where the existence of multiple disconnected canonical models implies that we must make a fundamental choice: in which domain we impose a distribution for the inflaton in order to then determine the likelihood of inflation.