Venator:
Gender-neutral toilets existed (unofficially) when I was at college back in 1973! The halls of residence, which had previously been 'single sex', became 'mixed'. The toilet facilities on alternate floors were labelled, by the college authorities' as 'Male' or 'Female'. It took us students only a matter of minutes to work out that it was far simpler to treat them all as 'unisex'. (The hall had previously been used solely by women, so there were only cubicles, not urinals). However we had to remember to tell our parents, when they came to visit, that (for example) they shouldn't be surprised when they met a fella' in the 'Ladies' ;-)
However 'neutral gender' toilets are now becoming standard in many places. Somerset House refurbished their public toilets last year so that everyone now uses the same extremely smart cubicles (which are all wheelchair accessible and with their own washbasins). I don't know why people should regard that as odd because many small restaurants (e.g. Indian restaurants) only have a single toilet and nobody objects to both male and female customers using the same facilities.
Chris