Why is it that in the morning when I turn on the lights, it is too bright to open both eyes at the same time, but I can open one eye at a time without much strain? It doesn’t matter which eye, and I can switch eyes quite easily. Is there some scientific explanation for this?
“Go get yourself some cheap sunglasses.” --Billy Gibbons
Excelent question JFMichael as I have often wondered this myself.
I could guess that keeping one eye closed tells the brain to adjust the iris of the other eye (make the hole smaller), as you are obviously in bright light (why else would you close one eye)
So here’s a question. If you have only one eye open when you walk into a brightly lit room from darkness, do both irises react to the light and constrict, or only the one that is being exposed to the light?
I have tried to experiment with this, but I need to open the closed eye to see the iris, and then it’s exposed to to the light. so that’s no good.
Well, a quick experiment on my part indicates that both of them probably react:
You need a dimly-lit room, a mirror, and a flashlight. Make sure that there’s enough ambient light to see your irises. Now shine the flashlight on one of your eyes and notice that both your pupils contract.
Half the pain.
Perhaps we are, unknowingly, “right-eyed” or “left-eyed”? Yes, both irises react, but perhaps one reacts a few microseconds faster? Just a WAG…
Either that, or it’s just the placebo effect! 
- Jinx
This is coming from a person who just had eye surgery about 12 hours ago, and is now wearing an eye patch on one eye: When you only have one eye open, different things happen. Your depth and focusing changes. My right eye has been covered all day (to the point where it doesn’t get any light, but it has enough room to open). Having only one eye open it is normally fine, but when I open the right one too, it’s painful. Anyways, I think that there might be another answer to the OP - I think it probably has something to do with the eyes working together to focus.
Here’s an applicable anecdote from my Army basic training days:
During our live-fire exercise which involved such fun activities as crawling through loops of concertina and dragging yourself through sand while live rounds are being fired overhead, there were also aerial flares being fired at regular intervals. While the flare was in the air we were instructed to not only stop moving, but to close one eye tight. When the flare went out, open the eye that was closed and you’ll still have decent night vision. It did seem to work.
Based on this, my guess is that the iris of the closed eye does not constrict (or constrict as much as the open eye). An easy experiment to try out would be to go outside your house/apartment/cave and look in the direction of a light source with one eye closed. Then turn off the light source, close the open eye and open the “protected” one and try navigating around in the dim light of the night. Then try the same experiment with both eyes open (and both eyes closed if you really must explore all the possibilities). Of course, street lights will ruin this experiment so you may have to cast about for a good place to try this out.