No, no, no! It's AC - Alternating Current. The polarity at the plug changes 50 times a second (actually it's 100, but that doesn't matter here). It's the magnetic field polarities between the rotor and field coil that determines the direction of rotation.
Your lathe is using a standard induction motor which will only run on AC. The direction of rotation is determined by a "magnetic kick" given by, usually, a start winding, which is then switched out of circuit. By reversing this winding, which is what the switch does, it kicks the rotor in the opposite direction. Once rotation is established it keeps going in the same direction until powered off.
As already mentioned, the motor you have is a Universal Motor. This will run from either AC or DC. The commutator, the segmented collar that the brushes run on, provides the reversal of the magnetic field to keep the motor spinning.
The magnet thing is the bit in the middle that gets fixed is it called a field winding, I pulled it out of the casing and reversed it and she still runs backwards