jdsgn: The deepest you can go into the 'whys' is starting off with gauge symmetries. I guess that doesn't really help.
The best reason I can intuitively give for the direction of the force a charged particle moving through a magnetic field experiences is accepting that a magnetic field cannot alter the kinetic energy of an electrically charged particle - thus acceleration must not have a component parallel or anti-parallel to the direction of movement. To change the kinetic energy, an object has to have a magnetic moment via which it couples to the field.
justanoldgamer: I almost understand this and I have a feeling I just need to work at it to get it. This post just became a candidate to be marked as solution.
After rereading your original post and doing some research on Faraday's motor, I guess I know what your problem is. If I'm wrong here, please supply some more information on the exact implementation of your motor.
You have a permanent magnet and a current carrying wire, both of which revolve with respect to each other.
In this case, the magnetic field caused by the electric current can be neglected. So, you have electric current from the wire and you have a permanent magnet with its own permanent magnetic field.
Now the wire is placed in such a way that it cuts the magnetic field lines in a right angle. Now charge traveling through the wire experiences a Lorentz force.