We address the subject of transport in one-dimensional ballistic polygon loops subject to Rashba spin-orbit coupling. We identify the role played by the polygon vertices in the accumulation of spin-related phases by studying interference effects as a function of the spin-orbit coupling strength. We find that the vertices act as strong spin-scattering centers that hinder the developing of Aharonov-Casher and Berry phases. In particular, we show that the oscillation frequency of the interference pattern can be doubled by modifying the shape of the loop from a square to a circle.