  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 $\alpha$-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 may reach 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 its late-time behaviour. If the evolution of the field is arrested
before this happens, however, we discover that quintessence can be achieved
without a potential offset. 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 $\alpha$-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.
