Speaking purely personally, I love a bit of old-fashioned chivalry from a gentleman. If he holds a door open for me I'll give him a smile and thank him and if he stands up when I come into the room he's the one I'll direct my greeting to.
It's not because I don't think I'm his equal. On the contrary I know damn well that, if anything, I'm his superior and he's acknowledging that by his deferrence. Women's Lib had some good points but, on the whole, I think it did women a great disservice because it made some women think they had to be on level terms with men. No way, Josephine, I'll stay above them, thanks.
It does no harm to work it the other way, either. I hate to see women who treat chivalrous gestures with contempt. To me, that's the height of ignorance and bad manners.