In old times, at least in Europe, discovering yourself was therefore a sign of respect, as it meant voluntarily leaving your guard down in front of someone.

In some Royal courts, that is actually how the king made you a noble, by just saying "you may cover yourself" - only the highest of social ranks was allowed to keep their hat on in front of royalty.

Here in Spain we learn about it because those tropes appear a lot in classic theatre: a person choosing to remain covered in a church is seen as blasfemy, not discovering your cape and hat while passing near a stranger at night is a sign that you are a criminal, etc.