Do electric cars have reverse gear or do they just invert current to make the engine spin in the opposite direction?
Depends on the car–both cases exist. The Tesla Model S, for instance, has only a single gear and hence uses the latter technique. In principle, it could go just as fast in reverse as it does forward 
(it’s electronically governed, of course).
Some cars have motors on the wheels, and don’t have any gearing at all. On the other hand, some conversion kits use the stock transmission on the original car and do have reverse and the other gears available.
Reverse the polarity!
Hey, it actually does have a real use!!
I’m sorry cap’n! I don’t have the power!
A traditional brushed permanent magnet (PM) DC motor will run backward when the polarity applied to the armature via the brushes is reversed. Wound field motors can be reversed by changing the polarity of either the field, or the armature, but changing both gives the same rotation. This is exploited in hand-held power tools which use series fields, and spin in one direction even though the AC power supply is constantly reversing.
But no modern electric cars would use power-wasting, high maintenance brush commutation. A modern car would use a brush-less DC or an inverter driven induction motor. These are fairly straight forward to provide reverse rotation, without requiring the high current switching of the older brushed motors…well not really, it’s just that the solid-state switches needed for reversing already exist in the drives required for these motors, and reversing just needs a bit more logic and maybe a second set of position sensing Hall devices.
The drive electronics can also do a fair job of simulating the action of a torque converter in an automatic transmission, and by switching various windings into series or parallel configurations, the motor can have a number of “gears”. A single mechanical reduction gear will most likely still be employed to allow a smaller, lighter motor to make the required power.
Just don’t cross the streams.