Personally, I do care how I am percieved. I want to be a legitimately good person. However, we sometimes do things that make it look like we don't care what others think of us because it doesn't always occur to us that what we are doing isn't normal. Even when beforehand we think of what a normal and appropriate thing to do might be we could totally miss the mark and do something weird that gives the impression that we don't care.
I know very closely several diagnozed asperger spectrum persons - I should have written their behaviour may seem non-caring when in fact they may try their hardest to behave as neurotypical would and to act in the best manners they understand. But human interactions are really complex and difficult.