Does the spin precession change sign when the angular momentum does?

Say you have a charged particle moving circularly in an electromagnetic field. Basic quantum mechanics tell us that its spin will precess with a certain frequency. If the same particle were traveling circularly in the opposite direction (e.g. clockwise instead of counterclockwise or vice versa) would the spin precess in the opposite direction as well, or would it stay the same?

EDIT: I'm assuming that in the second case, it's traveling in the opposite direction in the same field. This may mean that the field has the be purely electric, and not magnetic, for this scenario to be possible.

EDIT 2: If the field is purely electric, then classically the spin shouldn't precess at all. However, in the special relativistic case, the electric and magnetic fields are part of one tensor, and they both affect the precession. So if you're looking at this using a quantum mechanical Hamiltonian you may want to keep this in mind and use the Dirac equation instead of the Schrodinger equation.

-

The magnetic field $\vec B$ causes the spin-up and spin-down state rotate their quantum phases at different rates, because of the energy difference $-\vec B\cdot \Delta\vec \mu$ where $\vec\mu\sim \vec S$ is the magnetic moment proportional to the spin.
If the direction of the circle changes from clockwise to counter-clockwise, it indeed means that $\vec B$ had to change the sign, too. If it does, the difference between the rates of the spin-up and spin-down wave function changes, and the precession therefore proceeds in the opposite direction, too (assuming that the average component of the spin in the direction of the magnetic field – I mean in an axis whose direction we keep fix and don't reflect when $\vec B$ flips the sign – is conserved during the re-polarization of $\vec B$).
This isn't homework. Also, consider a particle moving at the correct speed for circular motion in a coulomb potential. Just apply the equation $F = \frac{mv^2}{r}$ and you'll see how a particle can move circularly in an electric field. –  Izzhov Jul 9 '13 at 15:35
All I'm asking is this: if you set a particle moving circularly clockwise in an electric field, the spin will precess in some way. To first order you can say that the spin is precessing around some vector $\vec{\omega}_s$. If you set the particle moving circularly counterclockwise in the same electric field, with otherwise all the same initial conditions, will that $\vec{\omega}_s$ point in the same direction (to first order) or the opposite diretion? –  Izzhov Jul 10 '13 at 16:19