I always thought he was a pretty ambiguous character in terms of gender, which is one reason why I like his char so much, he's not a cookie cutter type. I love Jack Sparrow as a character. Jack is wacky, spontaneous, witty, mysterious, etc, not to mention very attractive.
J. Depps characters Edward Scissorhands, Ichabod Crane, Willy Wonka and Jack Sparrow were all somewhat ambiguous in the gender category. I'm not sure if he does actually wish to play his roles like this or not, but there's no question that he is a very gifted actor.
Also when I found out what androgyne meant, he was one of the first (famous) people I thought of.
Off-topic- One of his children is named Jack

(Edit- But the name, and his daughter's name Lily might/probably be from 1985 film Legend, but, I don't know.)