Chivalry is not restricted to gender – it is a code of conduct that covers how to behave to one another. Ideally with honour and respect.
Am I more likely to hold the door for a woman than a man? No it really depends if someone is following me. However there are ways in which it may happen.
Today went to the aquarium in Plymouth with a zoologist. We were talking about kin relationships. I mentioned that I could see from birthdays I remembered and value of gifts given I was more disposed in giving to babies of female friends and female relatives as opposed to offspring of male.
Is this because I am certain of the mother in a way one cannot be of the father? I would like to think not consciously. “The Selfish Gene” is a good introduction to this topic that I can recommend reading.
So in a sense there could be a benevolent/altruistic sexism happening. Is this part of genetic disposition or cultural attitudes? It could just be good manners, and just maybe if it is genetic and helps your breeding chances it may still remain with us.