Swimming in pools is the same as swimming in water + bleach. It's the same chemical. I think that this is a point that often gets overlooked.
Swimming pools are chlorinated with calcium hypochlorite, which provides the exact same active ingredient as bleach, which is sodium hypochlorite. Notably, the concentration of hypochlorite is greater in bleach than in swimming pools. But, it's the same chemical. This is why the effects are the same. Let's list the two most obvious similarities:
Smell - Have you ever noticed how swimming pools smell like bleach? or that your body and swim gear smells like bleach after swimming?
Effect - Have you ever noticed how swimming pools tend to lighten (aka bleach) the colors from fabrics and hair? For example, you might notice that your color treated hair gets lighted by the swimming pool.
Pool chlorine and Bleach also react with your skin similarly. This is why it is very difficult to get the bleach smell off of your hands after working with bleach around the house. It's also very hard to get the bleach smell off of your hair and skin after you've gone for a swim.
What should you do? Not to sound like a broken record, but SwimSpray gets the chlorine off of you -- whether you've spilled bleach on your hands or you've just gotten out of the pool.