Before the Whitney Museum of American Art moved to its new location in Lower Manhattan, it hosted a discussion about what it means for a museum to be a safe and welcoming space.

Providing restrooms for everyone on the gender spectrum was near the top of the list.

“We invited artists of all gender identifications in,” said Danielle Linzer, the director of access and community programs, “and we heard loud and clear that it was something they really needed access to. Rather than being euphemistic, we decided to be direct.”

The signs at the new building say “All Gender Restroom,” and Ms. Linzer has observed women wondering aloud, “You mean I can go in the men’s room?”

The Whitney isn’t alone in being challenged to rethink one of the most basic uses of public space. With the issues of serving openly in the military and same-sex marriage now largely resolved, the fight for all-gender restrooms has emerged as the latest civil rights issue in the lesbian, gay, bisexual and transgender (L.G.B.T.) community — particularly the “T” part.