His face isn't a mask. He is the Joker and has no solid memory of who he used to be. He was originally a throw away character who was going to be in 2 or so episodes, then die. There is a possibility that he wasn't named, because the creators thought there was no need. Later he might not have been named due to tradition.
Wikipedia states "A definitive back-story has never been established for the Joker in the comics, and his real name has never been confirmed. He himself is confused as to what actually happened; as he says in The Killing Joke, "Sometimes I remember it one way, sometimes another... if I'm going to have a past, I prefer it to be multiple choice!" The first origin account, Detective Comics #168 (February 1951), shows the Joker had once been the criminal Red Hood. In the story, he is a chemical engineer looking to steal from the company that employs him. After committing the theft he is thwarted by Batman and falls into a vat of chemical waste. He emerges with bleached white skin, red lips, green hair and a persistent grin."