AFAIK that's a delay timer relay. Keith will be along to correct me shortly but I think when parked it checks around all the sensors every 6 minutes to see if the car is level. If it is not, if one corner is down for example, as it can't very well pump it up with the engine off it lets air out of the other 3 to level the car. This is why people with leaks will find their cars level but down on bump stops if they leave it overnight. A variation on the theme is where the car appears to have lowered slightly overnight but not to bumps and pops up once the engine is started. This will happen if its a very slow leak OR if you park on an incline (it just lowers the others enough to level the car) so bear that in mind! If you remove the relay when you park up for the night and there is a leak on one corner and your parking space is level, only the corner with the leak will be down in the morning which is a quick way to work out where to start looking. Do however read through Wammer's how-to, it will keep you from going in circles.
Also Keith you're right L-R readings across an axle are never identical but on mine they are as close as dammit. I've put a lot of work into the EAS over the last few years and am very pleased with it at the moment, it really does a good job when well maintained! As for Hawkeye, it allows you to force inputs, run the compressor, open and close valves and reset calibration to factory pre-set defaults. Keep in mind Hawkeye is made by the same guys that did Testbook and T4 which is the official dealer kit, its really not as bad people like to make out in here sometimes!