  Spin carriers propagating along quantum circuits gather quantum spin phases
depending on the circuit's size, shape, and spin-orbit coupling (SOC) strength.
These phases typically grow monotonically with the SOC strength, as found in
Rashba quantum wires and rings. In this work we show that the spin-phase
gathering can be engineered by geometric means, viz. by the geometric curvature
of the circuits, to be non-monotonic. We demonstrate this peculiar property by
using one-dimensional polygonal models where flat segments alternate with
highly curved vertices. The complex interplay between dynamic and geometric
spin-phase components -- triggered by a series of emergent spin degeneracy
points -- leads to bounded, global spin phases. Moreover, we show that the
particulars of the spin-phase gathering have observable consequences in the
Aharonov-Casher conductance of Rashba loops, a connection that passed unnoticed
in previous works.
