  We identify theoretically the geometric phases of the electrons' spin that
can be detected in measurements of charge and spin transport through
Aharonov-Bohm interferometers threaded by a magnetic flux $\Phi$ (in units of
the flux quantum) in which both the Rashba spin-orbit and Zeeman interactions
are active. We show that the combined effect of these two interactions is to
produce a $\sin(\Phi)$ [in addition to the usual $\cos(\Phi)$] dependence of
the magnetoconductance, whose amplitude is proportional to the Zeeman field.
Therefore the magnetoconductance, though an even function of the magnetic field
is not a periodic function of it, and the widely-used concept of a phase shift
in the Aharonov-Bohm oscillations, as indicated in previous work, is not
applicable. We find the directions of the spin-polarizations in the system, and
show that in general the spin currents are not conserved, implying the
generation of magnetization in the terminals attached to the interferometer.
