The new policy is a response to a student’s discomfort wearing a gender-specific uniform. It comes as Catholic institutions—like others around the world—grapple with society’s increasing questioning of what gender is. Innovations such as St. Brigid’s are pitting Catholic progressives against conservatives, who warn of spreading confusion about human nature.

“As a Roman Catholic school, our ethos would be based very much on respect and inclusiveness and reaching out with the hand of friendship,” said Tom Sherlock, chair of the management board at St. Brigid’s. “Society is changing and evolving all the time and this is a ...