A coil is symmetrical for electrical purposes, like a resistor, lightbulb, fuse, and other components. As long as black goes to one of the two terminals and the power to activate (switched by a relay) goes to the other side, it doesn't matter which wire goes to which side.
.................David Marston

Note on games this old sometimes it's stamped on the ends

Most coils in EM games, including yours, are running on AC voltage which means that the current through the coil and the magnetic field it creates in the coil change direction many times every second. But the steel plunger (or relay armature) doesn't care about the direction of the magnetic field. It's attracted to a magnetic field in any direction.

If the plunger was a magnet with a north and south pole instead of a steel rod, then the direction of the current and the magnetic field through the coil would matter and the plunger would be pushed back and forth as the direction of the current through the coil changed.

That's what makes a speaker (which uses a magnet) vibrate when an alternating current passes through the coil while a solenoid plunger just pulls in towards the magnetic field and stays there.

/Mark

Just a side note....it doesn't matter which wire goes to which side in your game...or in EM's running coils on AC, HOWEVER, it does matter how you wire up coils on newer machines (solid state) running DC power. The newer games have a diode across the coil connection and polarity is important when connecting these coils.

True BUT only because of the diode. The actual coil winding still doesn't care which way you connect the power to it - it will 'attract' the plunger regardless of the polarity.

AC is alternating current, a sine wave pattern. If the plunger was a permanent magnet you see some fun but its not. So the magnetic field is reversing 60 times a second but the direction of the pull remains the same was the temporary mag poles swap. EE 1980, a little of my college is left.