WASHINGTON — When last we saw the chief justice of the United States on the bench, John Roberts was joining with the Supreme Court’s liberals in an unlikely lineup that upheld President Barack Obama’s health care overhaul.

Progressives applauded Roberts’ statesmanship. Conservatives uttered cries of betrayal.

Now, the Supreme Court is embarking on a new term beginning Monday that could be as consequential as the last one, with the prospect for major rulings about affirmative action, gay marriage and voting rights.

Many people on both the left and right expect Roberts to return to the fold and side with the conservative justices in the new term’s big cases. If they’re right, the spotlight will be back on Justice Anthony Kennedy, whose vote typically is decisive in cases that otherwise split the court’s liberals and conservatives.

But Roberts will be watched closely, following his health care vote, for fresh signs that he’s becoming less ideologically predictable.

It may be that the dramatic health care decision presages “some shift in his tenure as chief justice,” said Steve Shapiro, the American Civil Liberties Union’s national legal director. “Or does it give him cover to continue to pursue a conservative agenda?”

The first piece of evidence could be in the court’s consideration of the University of Texas’ already limited use of race to help fill its incoming freshman classes, which comes before the court Oct. 10. The outcome could further limit or even end the use of racial preferences in college admissions.

Roberts has expressed contempt for the use of race in drawing legislative districts, calling it “a sordid business, this divvying us up by race,” and in assigning students to public schools, saying that “the way to stop discrimination on the basis of race is to stop discriminating on the basis of race.”

The written arguments submitted by both sides in the Texas case leave little doubt that Kennedy, not Roberts, holds the prized vote. The challengers of the Texas program and the university itself cite Kennedy’s prior writings on affirmative action a combined 50 times.

The court also is expected to confront gay marriage in some form. Several cases seek to guarantee federal benefits for legally married same-sex couples. A provision of the 1996 Defense of Marriage Act deprives same-sex couples of a range of federal benefits available to heterosexual couples.

Several federal courts have agreed that the provision of the law is unconstitutional, a situation that practically ensures that the high court will step in.

A separate appeal asks the justices to sustain California’s Proposition 8, the amendment to the state constitution that outlawed gay marriage in the nation’s largest state. Federal courts in California have struck down the amendment.

Once again, many legal analysts expect Roberts essentially to be against gay marriage. “The outcome clearly turns on how Anthony Kennedy votes,” said Georgetown University law professor Michael Seidman.

The justices may not even consider whether to hear the gay marriage issue until November.

Another hot topic with appeals pending before the high court, and more soon to follow, is the future of a cornerstone law of the civil rights movement.

In 2006, Congress overwhelmingly approved, and President George W. Bush signed, legislation extending for 25 more years a critical piece of the Voting Rights Act. It requires states and local governments with a history of racial and ethnic discrimination, mainly in the South, to get advance approval either from the Justice Department or the federal court in Washington before making any changes that affect elections.

The requirement currently applies to the states of Alabama, Alaska, Arizona, Georgia, Louisiana, Mississippi, South Carolina, Texas and Virginia. It also covers certain counties in California, Florida, New York, North Carolina and South Dakota, and some local jurisdictions in Michigan and New Hampshire. Coverage has been triggered by past discrimination not only against blacks, but also against American Indians, Asian-Americans, Alaskan Natives and Hispanics.

The court spoke skeptically about the provision in a 2009 decision, but left it mostly unchanged. Now, however, cases from Alabama, North Carolina, South Carolina and Texas could prompt the court to deal head on with the issue of advance approval. The South Carolina and Texas cases involve voter identification laws; a similar Indiana law was previously upheld by the court.

It is unclear when the justices will decide whether to hear arguments in those cases. Arguments themselves would not take place until next year.

Yet there still is a chance that the court could become enmeshed in election disputes, even before the ballots are counted. Suits in Ohio over early voting and provisional ballots appear the most likely to find their way to the justices before the Nov. 6 election, said Richard Hasen, an election law expert at the University of California at Irvine law school.