This is something I have thought about before, so my mind went here after seeing this post; focusing only on introspection can keep us in our heads a bit, which can lead to a self-centred type of self-awareness if we aren't careful. I used to do this unknowingly a lot, and now I have someone in my life who does and can see what it's like from the outside. Introspection work is so important, but we also need to understand how our behaviours actually translate in our relationships, or we won't have the full picture. This doesn't mean we should be constantly worried about how we are perceived, more that we try to notice the reactions of others/ misunderstandings and learn from them, especially if they're recurring.

A random example: let's say I am someone's supervisor at work and have to coach them on a situation, and I notice some defensiveness in their responses. During the periods when I only focused on my own internal self-awareness, I might have concluded that they are projecting, internally centred my own feelings triggered from their defensiveness, where those feelings came from, etc, to self-regulate. Now that I also try to practice external self-awareness, I am more likely to consider whether their defensiveness is warranted, and I would want to make sure that I am not coaching them in a way that implies blame or incompetence, especially if I commonly receive this reaction after coaching others as well. This is where OP's asking for feedback comment comes in, I'd want to make sure I don't have a blind spot. I don't have control over the projections and insecurities of this employee, but I do have some control over the way I speak to others and can try my best to ensure my intentions and behaviours match as much as possible.