Two somewhat contradictory sides of the new Supreme Court were on display Tuesday: The justices extended their efforts to maintain a low-key term mostly free of hot-button issues, while at the same time the court’s reinforced conservatives gave notice that big changes may be on the way.

After the bruising confirmation fight over President Trump’s nomination of Justice Brett M. Kavanaugh and an unwelcome turn in the partisan political spotlight, the justices largely avoided the high-profile controversies that have marked the court’s work in recent years.

On Tuesday, the justices delayed taking up a host of divisive issues, including immigration, abortion and discrimination on the basis of sexual orientation and gender identity. The court could still take up those issues, but unless it abandons its usual procedures, it is too late to hear them this term.

But meanwhile, the court’s conservative majority, fortified by Kavanaugh and Trump’s other nominee, Justice Neil M. Gorsuch, was at work.

It gave the Trump administration the green light to impose restrictions on transgender service members. The court answered the pleas of gun rights advocates by taking a case next term that could clamp down on local gun restrictions and give new definition to Second Amendment rights. And its four most conservative members — again, including Gorsuch and Kavanaugh — indicated they were eager to take on a case that could reexamine long-standing precedents and provide new protection for openly religious public employees.

In the push-and-pull of the day, the decision to take the gun case may have been the most telling about the court’s future.

“Nothing illustrates that change is on the way than the decision to take the Second Amendment case,” said Adam Winkler, a law professor at the University of California at Los Angeles who wrote “Gunfight: The Battle Over the Right to Bear Arms in America.”

The justices have not accepted a case involving gun restrictions since 2010, when they ruled that the Second Amendment right for individual gun ownership protected against state and local laws that would unduly limit it.

“The Supreme Court’s decision to hear this case sets the stage for affirming the individual right to self-defense outside of the home,” said Chris W. Cox, executive director of the National Rifle Association Institute for Legislative Action.

The speculation had been that neither conservatives nor liberals knew whether they could count on the vote of the court’s median justice, Anthony M. Kennedy. The court turned aside numerous challenges to restrictions on the kinds of guns that could be purchased, and whether they could be carried outside the home.

After one such decision in 2018, a frustrated Justice Clarence Thomas wrote that “as evidenced by our continued inaction in this area, the Second Amendment is a disfavored right in this court.”

But, Winkler said, “there is no hesitancy in Justice Kavanaugh,” who replaced Kennedy and has an expansive view of gun rights.

On the surface, the New York City law at issue might seem an odd choice for the next gun rights battle.

It is not replicated elsewhere, and it forbids the transport of even unloaded and locked-up handguns anywhere except to a firing range within the city. The challengers included licensed handgun owners who wanted to take their weapons to a firing range across the river, and to a second home upstate.

“The city does not even attempt to argue that what petitioners seek . . . poses any threat to public safety,” wrote former George W. Bush administration solicitor general Paul Clement, representing the petitioners. “The city’s ban cannot be reconciled with a Second Amendment that protects individual rights or with any meaningful level of constitutional scrutiny.”

Winkler said the case could be decided with a narrow ruling. But it is also “the first case on the right to have a gun outside the home,” he said, and could have great implications for how judges must consider challenges to laws in Washington, Los Angeles and other cities where there are rigorous — activists say onerous — regulations on carrying a weapon.

The court’s conservatives turned down the Trump administration’s attempt to have the court take up the challenges to the restrictions on transgender troops, where it generally has been on the losing side. But they granted the administration’s request to have the policy go into effect while the legal wrangling continued, most likely reaching the court next term.

“In declining to hear these cases, the Supreme Court saw through the administration’s contrived efforts to gin up a national crisis,” said GLAD Transgender Rights Project Director Jennifer Levi. “Unfortunately, the court’s stay of the lower courts’ preliminary orders means that courageous transgender service members will face discharges while challenges to the ban go forward.”

The court also took no action on the legal battle over the program that shields young undocumented immigrants brought to the United States as children from deportation, leaving in place the Obama-era initiative that the Trump administration has tried to end.

The justices would likely have had to act Tuesday to accept the administration’s request that it review the Deferred Action for Childhood Arrivals program, which has protected nearly 700,000 people brought to this country as children, commonly known as “dreamers.”

If the court sticks to its normal procedures, that would mean that even if it accepts the case at a later date, it would not be argued until the new term starting in October, with a decision likely in 2020. It is possible the justices saw no reason to take on DACA now, when it has emerged as an issue in the negotiations over ending the partial government shutdown.

Trump has offered extended but still temporary protection for those in the program in exchange for billions of dollars to build a border wall. Democrats have rejected the offer as “hostage-taking.”

If the issue is resolved by the political branches, the court might not have to get involved.

Other issues, however, might simply be a matter of time. The court still has not acted on Indiana’s recent restrictions on abortion, which were struck down by a lower court and are emblematic of moves by conservative states to restrict the procedure.

The case involves two provisions of a bill passed when Vice President Pence was the state’s governor in 2016. One provision requires that the “remains” of an abortion or miscarriage be buried or cremated, as required of other human remains. The other provision would prohibit doctors from performing an abortion if the woman is choosing the procedure because of the sex or race of the fetus, or because of a diagnosis or “potential diagnosis” of Down syndrome or “any other disability.” It requires doctors to inform the woman of the prohibitions.

Meanwhile, the Trump administration is an endless source of challenges for the court, where it believes its chances with the conservative majority are better than in the lower courts.

And Tuesday afternoon it was back with another request.

It involves whether the Commerce Department has the legal authority to add a citizenship question to the 2020 Census form. A federal district judge said Commerce Secretary Wilbur Ross violated a “veritable smorgasbord” of federal regulations by trying to add the question.

The administration asked the justices to bypass their normal procedures to hear the case this spring, because a final ruling is needed by June.

Legal experts say chances are good the justices will agree because of the unusual circumstances — and it shows that sometimes controversies find the Supreme Court, whether it wants them or not.