[inasp]

People broadly support transvestites and transsexuals not being discriminated against in scenarios like employment, access to housing, and so on. This approach to equality is analogous to what the gay rights movement achieved.

However the divisive issue is whether they should be treated as if they are the opposite sex in all contexts. This is a lot more unpopular, because it has the effect of harming and disadvantaging women and girls.

Turns out that most people don't appreciate the unilateral imposition of males on female-only spaces. Prisons, sports, locker/changing rooms, etc.