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I think this is interesting and likely this idea of reflection/mirrors is something to consider. Regarding his behavior after you challenged him to approach you - I'd be careful not to decide it's an ego thing and something to judge.
As a male who does go to the gym and do have some interactions at times, I am very conscious of the fact that women at the gym want to work out and if they're in their zone, I'll stay clear.
By ignoring him, you weren't projecting an openness and at the gym in particular, but always, I respect that. I had a woman introduce herself to me at a bar one night as someone who was in my 'workout class'. Since, we've chatted on and off (at gym and otherwise) but really I give her all the space.
I want to be friends with her and sometimes I've felt she was intentionally ignoring me but I don't push into that. Not because I'm afraid of being rejected but I'm confident and don't want to fuss with that. I don't judge her though. Things can be confusing in the matters. Lately, she started conversations with me I think because she noticed that I wasn't weird about things either way. I'm happy to engage with her and happy to not. These are relationships so, all parties need to be into it and you can feel it when someone isn't or, is trying to make you do something.
Be nice and open to him if you want to be. Even if he doesn't jump in right away, over time if you're meant to be friendly, you will be.
Do what you want to do is my opinion if that helps.
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