For skydiving instructor Donald Zarda, the termination came after the routine way he joked with a woman when the two had been strapped together shoulder-to-shoulder and hip-to-hip for a tandem jump. Something along the lines of, “Don’t worry, I’m gay.”

The three present the Supreme Court with a blockbuster question at the start of its new term: Is it legal to fire someone for being gay or transgender?

The cases will be argued Tuesday.

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This is one of the most consequential issues of the term, with more than 70 friend-of-the-court briefs dividing states, religious orders and members of Congress. More than 200 of the nation’s largest employers are supporting the workers.

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To some, it might seem surprising that this is an unsettled question. Cultural acceptance of the LGBTQ community is at a high. Transgender actors were spotlighted at the recent Emmy Awards, and “Orange Is the New Black” star Laverne Cox hit the red carpet accompanied by Chase Strangio, a transgender American Civil Liberties Union lawyer.

The gay rights movement has won victory after victory at the Supreme Court, capped by the landmark 2015 decision that established a constitutional right to same-sex marriage.

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But gay rights leaders say “married on Sunday, fired on Monday” is a possibility in more than half of the United States, where there is no specific protection for gay or transgender workers. The states that prohibit discrimination are not uniform — some protect only gender identity or transgender status, and some differentiate between public and private employment.

The issue for the court is the reach of Title VII of the Civil Rights Act of 1964, which, besides protecting against workplace discrimination because of race, also prohibits discrimination “because of sex.”

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For 50 years, courts read that to mean only that women could not be treated worse than men, and vice versa, not that discrimination on the basis of sex included LGBTQ individuals. The Trump administration says that is what the Supreme Court should find as well.

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“The ordinary meaning of ‘sex’ is biologically male or female; it does not include sexual orientation,” Solicitor General Noel J. Francisco said in a brief to the court in the Bostock and Zarda cases, which have been combined.

Employers violate Title VII only by treating members of one sex “worse than similarly situated members of the other sex. Discrimination on the basis of sexual orientation, standing alone, does not satisfy that standard,” he wrote.

Francisco makes a similar argument about gender identity in Stephens’s case.

That puts the Trump administration at odds with the Equal Employment Opportunity Commission, which decided in 2015 that gay and transgender individuals were federally protected.

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Treating a man who is attracted to men differently than a woman who is attracted to men is discrimination, the EEOC reasoned.

The commission also looked at a 1989 Supreme Court decision that said federal law protected against discrimination based on stereotypes; the court found for a woman who had not been promoted because her employers found her too aggressive, and her manner of dress not feminine enough.

Discrimination because of sexual orientation is the same thing, the EEOC said, because it relies on stereotypes about to whom men and women should be attracted.

Some courts have come around to those arguments. The full U.S. Court of Appeals for the 2nd Circuit ruled for Zarda, and said its contrary past decisions on the issue were wrong.

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Chief Judge Robert A. Katzmann wrote that “sexual orientation discrimination is motivated, at least in part, by sex and is thus a subset of sex discrimination.” (Zarda died in 2014, and his case is being advanced by his sister and partner.)

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The U.S. Court of Appeals for the 6th Circuit came to a similar conclusion in Stephens’s case. The panel found it “analytically impossible to fire an employee based on that employee’s status as a transgender person without being motivated, at least in part, by the employee’s sex.”

But in Bostock’s case, the U.S. Court of Appeals for the 11th Circuit went the other way, ruling for Clayton County, a suburb south of Atlanta, that Title VII did not protect on the basis of sexual orientation.

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Bostock ran a program for about a decade that recruited volunteers to serve as advocates for abused and neglected children. “We achieved great success, statewide recognition,” Bostock said in an interview.

“But everything changed when I joined a gay recreational softball league,” he said.

His job was recruiting volunteers, and he recruited from the league, too.

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One Monday in June 2013, Bostock found himself denied access to his office. He was told he was being fired.

The county says Bostock was terminated after an audit of the program, not because he is gay. Bostock’s suit was dismissed at an early stage because the lower courts ruled that Title VII did not cover sexual orientation.

There is little dispute about the facts in Stephens’s case.

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Stephens wrote an emotional letter to co-workers that said she suffered from gender dysphoria — a mismatch between the sex assigned at birth and a person’s gender identity — and said that after years of therapy she had decided to transition. Before surgical procedures, doctors said she should live for a year as a woman.

“At the end of my vacation on August 26, 2013, I will return to work as my true self . . . in appropriate business attire,” Stephens wrote.

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The funeral home’s owner, Thomas Rost, said he thought about it, but decided Stephens had to go. “He was no longer going to represent himself as a man. He wanted to dress as a woman,” Rost said in a deposition in the case.

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In a brief to the court, Rost’s lawyers elaborated: “He weighed the impact of his decision on Stephens and Stephens’s wife. He also thought about his female employees and clients who would be sharing a single-sex restroom with Stephens. Finally, he considered the impact on his clients’ grieving process. In the end, Tom could not agree to Stephens’s plan to violate the dress code, so he offered Stephens a severance.”

Harris Funeral Homes is represented by the conservative legal group Alliance Defending Freedom, which says Title VII was not violated. The funeral home would have fired “a female employee who insisted on dressing as a man while working with grieving families. . . . Because it does not disfavor one sex compared to the other, Harris does not discriminate based on sex,” the brief states.

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In previous rulings on gay rights, such as striking down a federal law that recognized only marriages between a man and a woman and state statutes that outlawed homosexual conduct, the Supreme Court relied on the Constitution.

These three cases concern the text of Title VII. No one argues that Congress in 1964 intended to protect LGBTQ individuals; homosexual conduct was illegal in the vast majority of the country. The solicitor general said that basic fact should decide the cases.

“In 1964, the ordinary public meaning of ‘sex’ was biological sex,” Francisco wrote. Congress is free to make clear that Title VII includes sexual orientation and transgender status, Francisco said, but it has repeatedly turned down attempts to change the law to do so.

The plaintiffs and their lawyers say that the statute’s text, not what Congress intended, should hold. That is something of a switch — usually it is conservatives making such a textualist argument — and the lawyers point to an unlikely ally: the late justice Antonin Scalia.

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In a 1998 ruling in a case called Oncale v. Sundowner Offshore Services, Scalia wrote for the court that male-on-male sexual harassment was covered by Title VII even though it is not mentioned in the statute or envisioned by Congress.

“It is ultimately the provisions of our laws rather than the principal concerns of our legislators by which we are governed,” Scalia wrote, adding that same-sex harassment need not be “the principal evil Congress was concerned with.”

“We’re the textualists here,” said Ria Tabacco Mar, an attorney for the ACLU, which is representing the Zarda estate. “We’re the ones pointing to the words of the statute.”

But besides Scalia, the court will be without former justice Anthony M. Kennedy, who had written the majority opinion in each of the gay rights movement’s victories at the court. Justice Brett M. Kavanaugh, who did not rule on major gay rights issues as a lower-court judge, is thought to be more conservative than his former boss Kennedy.

During a discussion of the case at Georgetown University Law Center, Supreme Court practitioner Roman Martinez said interpreting the text of the issue has scrambled usual alignments in the lower courts, with a leading liberal judge saying the statute does not cover sexual orientation and a leading conservative judge going the other way on transgender status.

It is possible the Supreme Court would rule differently on the two issues, and some observers think a prohibition on transgender status discrimination is closer to the statute’s wording, “because of sex.”

Nicole Saharsky, another lawyer who frequently argues before the court, thought that was more of a theoretical possibility than a real one.

“The Supreme Court’s going to say that there’s not a prohibition on discrimination on the basis of sexual orientation, but there is on the basis of transgender status?” she asked. “That just doesn’t seem like something they would do.”