Campbell is selling the notion that people of faith — not gay men and women — are the real victims. That’s a jarring claim to LGBT advocates, but it’s a tactic they acknowledge could upend the gay-rights debate. It’s not a new idea; claims of religious freedom have been used for decades to stave off the reach of nondiscrimination laws. But it’s one that opponents of gay rights hope resonates with the public — and perhaps, down the road, the courts. They feel they have found a higher ground in a debate they otherwise appeared destined to lose. “I think there is an opportunity for our side if we can help people understand how this movement is not, or isn’t anymore, about obtaining freedom for homosexuals,” said Peter Sprigg, senior fellow for policy studies at the Family Research Council. “It’s about taking freedom away from anyone who disagrees with them or their conduct. I do think that goes beyond what most Americans support.”

The Huguenins’ case isn’t an isolated one. Campbell, who is part of the Christian nonprofit group Alliance Defending Freedom, said he is working on similar cases in Lexington, Kentucky, where a T-shirt printer refused to make apparel for a gay-pride celebration, and in Washington state, where a florist declined to make floral arrangements for a gay-marriage ceremony.

Beyond those cases, the issue may move to the forefront of the public’s mind when the Senate begins debate this fall on the Employment Non-Discrimination Act, which would ban discrimination on the basis of sexual orientation in hiring and firing employees. The act, known as ENDA, is different because it’s about employment rather than public accommodation, but it raises similar questions of religious liberty — whether, for example, a private Christian bookstore can refuse to hire a gay clerk. (Gays and lesbians are not considered a protected group under the landmark Civil Rights Act of 1964.)

Gay-rights supporters acknowledge the contours of the fight will be different — and likely more difficult — than the battle over marriage equality. An argument focused on religious liberty, they say, is a potent one. “That’s absolutely now the primary argument opponents of LGBT equality are making,” said Brian Moulton, legal director of the Human Rights Campaign. “They’re trying to assert that the people experiencing discrimination here aren’t LGBT people necessarily but are people of faith who want to preserve their opposition to homosexuality.”

Polling suggests that Moulton shouldn’t worry. A survey conducted earlier this year by the Human Rights Campaign and the Third Way, a centrist Democratic think tank, found broad opposition to new laws that would allow businesses to deny services to gay men and women. Sixty-four percent, for instance, said they oppose such a law even if it only applied to small businesses providing wedding-related services. In a question tailored for the New Mexico case, 54 percent of adults said it was wrong for a business to deny wedding-related services to a gay couple; only 15 percent agreed the company should be allowed to refuse service for weddings because of religious reasons.