Most single-phase motors have at least two windings (coils). Notable exceptions are the shaded-pole type (usually found in bathroom fans and other ultra-low HP applications), and the universal type (the type with brushes, usually found in portable power tools, kitchen appliances, etc.).
The motor in question here is an induction-run. It is either repulsion-start, capacitor start, or split-phase start.
In an induction motor, there is one run winding, and one start winding. If the motor is dual voltage, there are two run windings, and one or two start windings. The run windings are bi-directional. This means that they will turn the shaft in either direction. When the shaft is at a standstill, they don't know which direction to go though.
This is where the start winding comes into play. It is slightly out of phase with the run winding, and will cause the shaft to turn in one direction only, depending on whether the phase relationship is clockwise or counterclockwise. In a capacitor-start motor, the capacitor in series with the start winding causes this phase shift. In a split-phase motor, the start winding is mechanically shifted on the stator. In a repulsion-start motor, the rotor is wound, and the brushes cause an electrical phase shift. In all of these motors, the start winding is energized only during starting. It is disengaged at about 2/3 speed.
For a capacitor-start or split phase motor to be electrically reversible, it needs at least 4 leads out if it is single voltage. If it is dual voltage it needs at least 6 leads out. A repulsion-start motor is not electrically reversible at all, the position of the brushes is mechanically changed to reverse the shaft rotation.
If a motor is dual voltage, and has 4 leads out, it is either not reversible, or it is repulsion-start. If it is capacitor-start or split-phase, it can be reversed only by internal re-connection of the start winding. Reconnecting the 4 external leads will cause the two run windings to fight each other, and the shaft won't turn at all.