Removed hats are held in such a way that only the outside and never the lining is visible.
Hats are removed when inside,except for places that are akin to public streets,like lobbies, corridors, and crowded elevators. The elevator is considered a public place.
You may choose to remove your hat in a public elevator,but in the presence of a lady,your hat must be removed.
Tipping of the hat is a conventional gesture of politeness.
This custom has the same origin as military saluting,which came from the raising of medieval knights visors,to show friendliness.
At the turn of the century,all adults wore hats whenever they left the house.
It was a matter of good personal hygiene since hats were protection from industrial dirt.
Why are there different rules for men and women?
It may have to do with the difference in the styles of men's and women's hats.
Men's hats are easily removed,but women's hats with ribbons,bows,flowers and other decorations can be quite a production to remove,especially if they are anchored with hat pins. Women might also risk messing up their hairdos if they had to remove their hats.
A lady,however,never wore a brimmed hat after 5pm,a fashion rule that developed because she didn't need a brim after sunset.