Since being anonymous means you don’t use body parts to back up your words, let me just say you’re wrong. That hat etiquette for women is different and women do not remove outfit- specific hats or scarves worn as accessories.
Not in church or funerals or ceremonies etc. a Kentucky derby hat, the big church hats Easter hats, cocktail fascinators, the pillbox with veil for funerals these are all outfit hats and don’t remove indoors.
Now let’s do you- which way do your cummerbund pleats face?