What destroys a LED in the reverse direction?
ESD appears to cause damage due to hot spots or some other localized damage. I have seen heterojunction LED failures that appear to be partial. 

Failure with DC current in reverse is probably related to power dissipation, but it might be unwise to depend on it. Breakdown could be quite high, hence the allowable current could be quite low (maybe less than 1mA). 

Safest is to follow the LED data sheet recommendation- usually 5V or so reverse is guaranteed. Many types of LEDs have much higher actual reverse voltage breakdown (perhaps 15V to 70V), but it's unwise to depend on it- the LED maker could change the chip supplier or process or purchasing could go to a different vendor. 

The typical situation where LEDs are exposed to reverse voltage is when they are operated in a multiplexed configuration- they will see up to the supply voltage in reverse. It's not really a good idea for efficiency to make the supply voltage much higher than the sum of the series LED forward voltages (often, but not always, just one LED is used). For example, you might use 5V for a single 2-3V LED array, or 12V for an array of series strings with 6-9V per string. Since the individual LEDs can take 5V each (usually guaranteed), you'd be fine in either case.

See this nice instructables gif: