I'm not sure if it's a portrayal of a "gender-neutral" character as such but It's still quite interesting. In Resident Evil: Code Veronica X, two of the villans you meet are Alfred Ashford and his Twin sister Alexia, the grandchildren of the Umbrella co-founder Edward Ashford. But it turns out later on in the game that Alexia was missing (presumed dead) and that in her absence Alfred because pretty much lost his mind and started imitating his sister, wearing dresses, changing his voice and it actually took me by surprise. Until it was revealed I genuinely thought they were two different people.
One character who has been the subject of much debate around the issue is Birdo from Mario 2, who might be the earliest example of a trans character in a game. It all comes form the original English manual, but references to this have been removed in later releases like Super Smash Brothers:
Well like it or not creatures that are "genderless" or "androgynous" is a thing that legitimitly exists in nature so take it up with Gaia. I can think of a few example's in fantasy and science fiction as well.