  We analyze and compare the multi-field dynamics during inflation and
preheating in symmetric and asymmetric models of $\alpha$-attractors,
characterized by a hyperbolic field-space manifold. We show that the
generalized (asymmetric) E- and (symmetric) T-models exhibit identical
two-field dynamics during inflation for a wide range of initial conditions. The
resulting motion can be decomposed in two approximately single-field segments
connected by a sharp turn in field-space. The details of preheating can
nevertheless be different. For the T-model one main mass-scale dominates the
evolution of fluctuations of the spectator field, whereas for the E-model, a
competing mass-scale emerges due to the steepness of the potential away from
the inflationary plateau, leading to different contributions to parametric
resonance for small and large wave-numbers. Our linear multi-field analysis of
fluctuations indicates that for highly curved manifolds, both the E- and
T-models preheat almost instantaneously. For massless fields this is always due
to efficient tachyonic amplification of the spectator field, making
single-field results inaccurate. Interestingly, there is a parameter window
corresponding to $r={\cal O}(10^{-5})$ and massive fields, where the preheating
behavior is qualitatively and quantitatively different for symmetric and
asymmetric potentials. In that case, the E-model can completely preheat due to
self-resonance for values of the curvature where preheating in the T-model is
inefficient. This provides a first distinguishing feature between models that
otherwise behave identically, both at the single-field and multi-field level.
Finally, we discuss how one can describe multi-field preheating on a hyperbolic
manifold by identifying the relevant mass-scales that control the growth of
inflaton and spectator fluctuations, which can be applied to any
$\alpha$-attractor model and beyond.
