You can open your eyes without discomfort because of 2 things:
1) The salinity of the pool water closely matches human tears so it's not irritating.
2) Your SWG is keeping a consistent level of chlorine in your pool to ensure that any irritating agents are rapidly metabolized.

Chlorine is chlorine. Your SWG makes chlorine by breaking down sodium chloride to release the chlorine. When it metabolizes something (algae, bacteria, suntan lotion, etc) it recombines back into sodium chloride. (I don't really understand this part of the chemistry--Chem_Geek will probably be happy to explain it if you need to know).

Of course, sodium chloride's common name is salt.

Most people's eyes are irritated by improper pool maintenance. Free Chlorine is not sufficiently maintained and contaminants appear, particularly Combined Chloramines.

CCs in your water mean there is some contaminant that your Free Chlorine is metabolizing.
CC gives pools that characteristic unpleasant "Oh there's too much chlorine" smell.
CC irritates your eyes and skin far more than Free Chlorine.

My pool and the pools of Watermom, Poconos and Aylad ALL have fairly high Free Chlorine levels (all easily exceed the absurd max recommended by the Government of 3ppm), yet rarely do ANY of us have that chlorine smell, nor is our water particularly irritating.

I inadvertently (don't ask!) put 160 lbs of sea-salt in my water and it is less irritating because it is closer to human tears than it was, but that's all.

So, you didn't switch from "chlorine" to "SWG". Both are chlorine pools, and, if necessary (and probably occasionally) you can add bleach to your pool.

And don't blame the chlorine for the irritated eyes!

Meanwhile, enjoy your sanitary, non-irritating pool!