I always wished I had an invisibility cloak - since years before Harry Potter came along - for those times when I'd rather not face the world at all, or rather, the people in the world. It would be wonderful to have that option, to vanish, to observe unseen, to walk around without being noticed.
Avoiding people's eyes is a subconscious way of attempting to vanish. I do that a lot, even now. It's like a defiant 'I don't want you to see me - I just want to be me, without you looking'. Shame it's an illusion.
I sometimes think I'd make a better spirit guide than I would a physical person. I could appear in a form pleasing to the person, whisper words of encouragement or prophecy, then vanish away. That would work for me. I'd have the interaction I needed, on my terms, be a useful part of existence, and be in control of how I appeared to others.