Electron spins do not play a role in the geomagnetic field generation. The temperature of the core is >5500 K, well above the ordering temperature of Fe (the primary constituent of the core). This means that thermal fluctuations are more than sufficient to destroy any ordering of the spins.
The geomagnetic field is generated by the fluid motion of the electrically conductive fluid outer core around the solid inner core. The fluid motion of the outer core is driven by both thermal and compositional convection (as Fe freezes out of the liquid lighter elements remain creating compositional buoyancy) and is dominated by large scale flow. The Earth’s rotation also plays a big role; it produces convection columns within the outer core that align along the rotation axis. So to answer your question, thermal and compositional convection along with the Earth’s rotation control fluid flow in the outer core. Why one way or the other? The flow of fluid is approximately axial-symmetric. The thermal and compositional convection is radial and the Earth’s rotation adds a helical twist to the fluid motion.
I guess what you are really asking is why does the geomagnetic field point north or south? First some background. As I mentioned the outer core is conductive and, in the presence of a magnetic field, electric currents will be produce inducing new magnetic. This is the basic premise of the self-sustaining dynamo, which is a big feedback system of convection to electric currents to magnetic fields, which then modify convection currents (through magnetohydrodynamics;
http://en.wikipedia.org/wiki/Magnetohydrodynamics ), and so on.
Now, imagine you have a convecting Earth-like core system in the absence of any magnetic field, i.e., convection without magnetic induction. Then a seed magnetic field was instantaneous “switched” on; the final stable configuration would depend on the interplay of the strength and direction of the seed field, the configuration of convection before the seed field (and how it changed through time, i.e., magnetohydrodynamics), magnetic diffusion through the solid inner core and its relative scale to that in the outer core (the inner core can act as a breaking system to changes in the magnetic field induced by convection in the outer core), controls on the heat flux through the core mantle boundary, and a host of other details. So those are (some of) the factors that control the orientation of the magnetic field.
And to answer an inevitable question, no, we don’t know what the source of the geodynamo seed field was.
I hope that goes some way to answering your question.
