Two unknowns - whether the court will be drawn into legal controversies arising from the House Democrat's impeachment inquiry into President Donald Trump, and the health of the court's oldest member, Justice Ruth Bader Ginsburg - add to the uncertainty.
Resolution of the most contentious cases could happen in June, in the heat of a presidential campaign in which the future of the court has emerged as a galvanizing issue for conservatives and liberals.
On the court's agenda:
• Whether federal law protects LGBTQ workers from discrimination or being fired.
• If the Trump administration's efforts to end the Obama-era program that protects immigrants brought to this country as children are lawful.
• The first Second Amendment claim involving gun ownership in more than a decade.
• Whether a state may withhold aid to private religious schools if it offers funding to secular ones.
• An abortion case that gives the court's new conservatives an opportunity to begin reconstructing its jurisprudence on what is perhaps the nation's most divisive subject.
On the horizon, there are cases that could re-define when the government must give greater deference to a person's religious beliefs, and perhaps even a third trip to the high court for the Affordable Care Act.
Last term, after the partisan bitterness that accompanied Justice Brett Kavanaugh's ascension to the court, the justices sought common ground on some issues and put off others - abortion restrictions, for instance, and Barack Obama's Deferred Action for Childhood Arrivals program.
The delay is over, and the conservative majority - bolstered by Kavanaugh and Justice Neil Gorsuch, Trump's other appointee - is in position to be more assertive this term, according to those who watch the court.
"Probably not the revolution that some seek and others fear, but we will likely see a court moving further and faster in a rightward direction," said Irv Gornstein, executive director of the Georgetown Law Center's Supreme Court Institute. "The docket almost guarantees it."
Some conservatives see opportunity.
"I actually can't recall a time in the last 20 years that there were this many key issues that seemed ready for decision and primed for decision, and a court that seems open to them," said Mark Rienzi, president of the Becket Fund for Religious Liberty.
Determining how far and how fast the court moves is Roberts, 64, entering his 15th year as chief justice and his second as the court's pivotal member.
Since the retirement of Justice Anthony M. Kennedy in 2018, Roberts is the median between four more conservative colleagues on one side and four liberals on the other.
He uses every public appearance to try to persuade that the court - with conservatives chosen by Republican presidents and liberals by Democratic ones - may be ideological but is not partisan.
"When you live in a politically polarized environment, people tend to see everything in those terms," Roberts said last month at an event at New York's Temple Emanu-El. "That's not how we at the court function and the results in our cases do not suggest otherwise."
Roberts has noted the range of majority lineups last term, even in cases that divided 5 to 4. The court's dominant conservatives made up the majority in only about one-third.
In the last term's two most important cases - whether federal courts have a role in policing electoral maps for extreme partisan gerrymandering and whether the Trump administration could add a citizenship question to the 2020 Census forms - Roberts gave his first glimpses of how he might play his pivotal role.
He voted no in the gerrymandering case, joined by the conservatives, and no in the Census case, joined by the liberals.
At the event in Manhattan, Roberts lamented that the discord between the legislative and executive branches leaves the court with more decisions to make.
"We do seem to be getting more and more involved in every aspect of society in a way that would have been surprising to the framers of the Constitution," Roberts said at one point.
At a Supreme Court preview at George Washington University's law school, Sherrilyn Ifill, president and director-counsel of the NAACP Legal Defense and Educational Fund noted that the court remains "the most powerful body in this country."
David Cole, legal director of the American Civil Liberties Union, said that last term "the court seemed to have done everything it could to rise above the partisan rancor to not be divided along partisan lines in the ways so much else is in our society." He added: "This term it's going to be harder."
The court's decisions will impact 800,000 "dreamers," in the DACA case, Cole noted, millions of LGBTQ workers in deciding whether federal discrimination laws protect on the basis of gender identity and sexual orientation, and "half the country" in the abortion case.
"It will be a momentous term for the rights of individuals," Cole said.
Some conservatives say the court's push to prove its nonpartisanship is a one-way street: that the expectation is one of the justices on the right must bend to join the liberals, who vote more consistently as a group.
"I'm concerned about a pattern of increasing threats to judicial independence," said Carrie Severino of the Judicial Crisis Network. She and other conservatives are outraged by a brief filed by Sen. Sheldon Whitehouse, D-R.I., and four other Democratic senators in the gun case that warned the court in stark terms that it is perceived as partisan and beholden to conservative groups.
Severino called it a "thinly veiled ultimatum to Chief Justice Roberts ... suggesting that if the Court didn't rule as (Whitehouse) requested that he would proceed with legislation to pack the court."
The abortion case the court accepted Friday puts Roberts in a particularly bright spotlight.
The court said it will review a Louisiana law that requires doctors performing abortions to have admitting privileges at nearby hospitals. It is almost identical to a Texas law that the court struck down in 2016, saying the requirement was meant more to hobble abortion clinics and restrict women's access to the procedure than to protect their health.
Roberts was a dissenter in the Texas case, but he joined the liberals in February to keep the Louisiana law from going into effect after a lower court approved it.
"It will reveal probably more than any case this term this emerging role of Roberts as the swing vote," said Jonathan Turley, a GWU law professor.
Some events are beyond the court's control.
Ginsburg's health is one of them. Last December, the senior liberal member of the court had part of one lung removed after cancer was discovered. The recovery caused her to miss a round of oral arguments for the first time in 26 years on the bench.
This summer, the 86-year-old announced she had undergone radiation treatment for a tumor on her pancreas.
She has counteracted the worries about her condition with an impressive show of vigor: nearly a dozen speaking engagements over the last month-plus, traveling to Buffalo, Chicago, Little Rock, New York and elsewhere.
But if her health forced her from her seat, the Republican Senate has made clear it would move quickly to ensure Trump would name her replacement and solidify conservative control of the court.
Another potential controversy for the court: the impeachment inquiry. Nothing would put the court more in the political spotlight than having to rule on issues relating to the proceedings against the president.
But lawsuits seeking Trump's tax returns and alleging he's violating the Constitution with transactions through his family business with foreign governments are making their way through the legal process.
Congressional subpoenas for information from the executive branch are being ignored, and some on the left say the judiciary has a role to play in enforcing them.
"Where the House is seeking information that is relevant to its determination whether to impeach the president, it needs that information as quickly as possible," said Brianne Gorod, chief counsel of the liberal Constitutional Accountability Center.
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