Visitors line up to enter the Supreme Court on Tuesday. (Win Mcnamee/Getty Images)

The reality of an ideologically divided, evenly split, one-man-down Supreme Court became apparent Tuesday: The justices deadlocked on a major organized-labor case and tried to avoid a second stalemate by floating their own policy compromise on the Affordable Care Act’s contraception mandate.

The actions underscored how the death of Justice Antonin Scalia has upended business as usual at the Supreme Court. They also provided a preview of the kind of tentative decisions that may be all the eight-member court can muster as it faces a docket filled with election-year controversies.

The new order demanding additional briefing in the contraceptive cases, which were argued before the justices last week, was the most unexpected and unusual of the actions. Even more surprising, the court asked the parties to react to a compromise it created.

Calling for the Obama administration on one side, and the objecting religious charities, hospitals and universities on the other, to weigh in on such a specific proposal crafted by the court is an indication that it is ready to undertake creative moves to avoid a series of 4-to-4 votes.

But the court’s actions seemed to reinforce another deadlock — over whether the Republican Senate leadership should move forward on considering President Obama’s nominee for Scalia’s seat, Judge Merrick Garland.

Sen. Patrick J. Leahy (D-Vt.), who is leading the Senate effort to force hearings on Garland, said, “When the Supreme Court is evenly split, like today, it cannot perform its function under the Constitution to serve the people.”

Conservatives responded that a divided court is better than one with five left-leaning justices.

“If President Obama were to succeed in shifting the Supreme Court dramatically to the left, with the Senate confirmation or recess appointment of Merrick Garland or any other liberal, the court would become a rubber stamp not just for the wishes of powerful labor unions, but also for virtually the entire progressive agenda,” said Curt Levey, the executive director of FreedomWorks Foundation, which is lobbying senators not to vote on Garland.

Tuesday’s deadlock in the union case was a win for liberals.

The justices said they were split on a challenge brought by a group of California teachers — and backed by conservative legal groups — who say that their free speech rights are violated when they are forced to pay dues to the state’s teachers union.

Without Scalia, there were not five votes for overturning a Supreme Court precedent, more than 40 years old, that held the opposite and said unions can collect fees from nonmembers to cover the costs of collective bargaining. More than 20 states allow the fees, and unions said prohibiting them would hobble pub­lic-employee unions, the healthiest segment of the labor movement.

The 4-to-4 vote maintains the national status quo.

The case showed the difficulty conservatives will face in putting together a five-member majority when the court’s four liberals stick together, as they have in the past.

But not all split decisions this term would benefit the left.

A tied vote in an upcoming case on Obama’s plan to shield millions of illegal immigrants from deportation would mean the program could not commence while he is in office. A lower court blocked it, and a split on the Supreme Court would uphold that decision.

The court faces some urgency in trying to find a solution on the contraceptive cases. Eight of the nation’s regional appeals courts have upheld the mandate, while one has struck it down. Simply reaffirming those decisions would mean that the national health-care law would be enforced differently depending on the location of an organization and its employees.

At last week’s hearing, the court’s four liberals seemed to agree that the Obama administration had offered an acceptable compromise for religiously affiliated organizations such as universities, hospitals and charities that want to be freed from the obligation to supply their female employees with no-cost contraceptive coverage, which the organizations say violates their religious beliefs.

The accommodation requires the groups to tell the government they object, then allows the government to work with the groups’ insurers to provide the coverage without the organization’s involvement or financial support. Insurance companies say providing birth control is cost effective, and the government subsidizes it in some cases.

But the justice who could provide a fifth vote, Justice Anthony M. Kennedy, agreed with the court’s other conservatives that it sounded as though the government was “hijacking” the insurance plans to provide contraceptive coverage, rather than finding a way to provide the coverage without involving the groups.

In the order issued Tuesday afternoon, the court directed both sides to file briefs that address how employees could receive contraceptive coverage through their employers’ plans, “but in a way that does not require any involvement” from the employers beyond their decision to provide health insurance.

The court went into unusual specificity in asking the parties to address how that could happen, and it outlined a scenario.

It said an organization could contract to provide health insurance for its employees but inform the insurance company that it did not want the plan to include contraceptive coverage that it found objectionable.

Then the insurer could separately notify employees “that the insurance company will provide cost-free contraceptive coverage, and that such coverage is not paid for by petitioners and is not provided through petitioners’ health plan.”

Under such a plan, an organization would not have to submit a form to the government or its insurance company addressing the coverage.

Lawyers on both sides were initially positive.

“This is an excellent development,” said Mark Rienzi, lead attorney for the Becket Fund for Religious Liberty, which is representing a group of nuns called the Little Sisters of the Poor. The nuns say providing employees with contraceptives would violate their faith.

“We look forward to offering alternatives that protect the Little Sisters’ religious liberty while allowing the government to meet its stated goals.”

Women’s groups disheartened by last week’s hearing welcomed the court’s attempt to find a compromise.

“It’s a positive sign that the order suggests the justices have moved beyond the unrealistic alternatives and are focusing on women getting contraceptive coverage from their regular insurance company,” said Gretchen Borchelt, vice president for reproductive rights and health at the National Women’s Law Center.

She said it was not clear to her that the groups or the government would agree to the compromise, but “our bottom line is that the women get seamless contraceptive coverage from the same insurance company.”

In its order, the Supreme Court said that the parties “may address other proposals along similar lines” and that the briefing should be completed by April 20. It did not indicate whether it would schedule a rehearing.