There are so many possible causes of this...you mentioned adrenals, thyroid, something glandular, possibly mineral imbalance, etc...others have suggested to me in the past that it may relate to the metals or toxin load in the body, and certainly in my case the diabetes coupled with being overweight can contribute.
I've gotten used to living with it when it happens, and I certainly identify with the salty sweat in the eyes. I have to be careful because I can taste my sweat within 2 to 3 hours of working and realize that there is no salt left. That's when I head inside the house and start hitting the minerals, potassium supplement, water, admittedly sometimes even potato chips, to get some salt/sodium into my system again. One person told me that you can get a day's requirement of sodium by rinsing and spitting water with baking soda in it, which is also supposed to be a good way to neutralize the acid balance in the mouth, but I've never read anything about it.
I remember trying to sell Avon to young college girls years ago (in another lifetime, it seems), and realizing they were absolutely fascinated by watching the streams of water running down my face and dripping onto my shirt (and Avon products, if I wasn't careful) as I am trying to convince them to buy cosmetics. While fascinating, I'm not sure this is a particularly attractive trait...at least I know my pores are open and working! I still turn down the rare invitations to participate in sweat lodges, with great respect. I am a walking sweat lodge unto myself at times and don't need any assistance.