A DC motor has a permanent magnetic field created by a magnet, and is reversed by reversing the polarity going to the brushes. An AC motor has windings in place of that magnet, and is reversed by keeping the polarity of the brushes (or field) the same while reversing the polarity of the other. So when an AC motor is reversed one input remains the same while the other is reversed, allowing it to swap directions because the brushes and field are now in opposite phases.
Yes, an AC motor works on DC just fine, however reversing the polarity won't reverse the motor.