When you're in Rome, should you always do what the Romans do?

A respectful person, I believe, will learn the symbols of politeness in the place they are visiting or moving to, and try to adopt them. To not adopt the local culture, seems to me to be a sign of disrespect.

But there, things get complicated. If adopting the local culture means giving up your birth culture, you would be disrespecting your ancestors. Which do you do? It seems like an impossible dilemma or boundary to straddle, and is probably what leads to so much of the discord between local people and immigrants.

Just because it is polite to act as your hosts do, do hosts have a right to expect visitors to act as they do?