WASHINGTON — The U.S. Supreme Court term that begins next week will test the willingness of Chief Justice Roberts and Justice Alito to stand by previous high-court rulings that bolstered affirmative action and abortion rights.

Justice Roberts, 51, and Justice Alito, 56, pledged after they were nominated last year that they would respect past decisions. Justice Roberts warned in Senate testimony of a “jolt to the legal system” when the high court reverses itself.

Conservative groups nonetheless are asking the two new justices to try to push the court to roll back precedents in both areas of law. The challenge is most direct in a fight over a federal ban on late-term abortions. The high court in 2000 struck down an almost identical Nebraska law.

How the court deals with the 2000 precedent will “tell us a little bit about the soul of the Justice Roberts court,” the New York-based legal director of the American Civil Liberties Union, Steven Shapiro, said. Mr. Shapiro’s group opposes the abortion law. The case is set for argument November 8.

The nine-month term also features two challenges to the use of race in assigning students to public schools. Those cases raise questions about a 2003 Supreme Court ruling that let universities consider race as an admissions factor.

Other top cases on the court’s docket include fights over punitive damages and federal regulation of auto and power-plant emissions. The high court added nine new cases to its calendar Tuesday, including fights affecting the insurance and waste-disposal industries.

Some backers of the federal abortion law have said the dispute presents an especially strong case for overturning a precedent. The 2000 ruling, Stenberg v. Carhart, said the Nebraska statute was too broad and should have allowed an exception to protect the mother’s health.

The 5–4 ruling hasn’t had much impact on American society or the law, unlike, for example, the 1966 Miranda ruling requiring police to tell suspects in custody of their right to remain silent, a law professor at the University of Notre Dame in South Bend, Indiana, Richard Garnett, said.

“They could overrule it and still maintain plausibly that the role of judges is supposed to be humble and restrained and respectful of precedent,” Mr. Garnett said.

The Bush administration, saying the federal and state laws define the banned procedure differently, contends the court can uphold the law without reversing Stenberg. At the same time, the government says the Stenberg ruling should be overturned if the court concludes the two laws can’t be distinguished.

That would mark a major statement about the direction of the Roberts court, according to a law professor at Georgetown University in Washington, Susan Low Bloch.

“If they overturn a precedent that’s only six years old in the abortion context, that’s going to send out a lot of signals to a lot of people,” Ms. Bloch said.

Stenberg was a 5–4 decision, with now-retired Justice O’Connor in the majority and the late Chief Justice Rehnquist in dissent. To overturn it, both Justices Roberts and Alito would have to join the majority, as would Justice Kennedy, a swing vote on abortion cases.

Justice Kennedy co-wrote the 1992 Planned Parenthood v. Casey decision that reaffirmed abortion rights. The ruling relied heavily on stare decisis, the notion that justices should be slow to overturn settled precedents. Justice Kennedy switched sides in Stenberg, complaining in dissent that the majority had gone too far and effectively barred any state regulation.

The education cases concern efforts by school districts in Seattle and Louisville, Kentucky, to promote racial diversity. A key precedent will be the court’s 2003 Grutter v. Bollinger ruling, which let race be considered in university admissions.

Although the families challenging the Seattle and Louisville programs aren’t seeking to revisit the 2003 ruling, Governor Bush of Florida and others have filed friend-of-the-court briefs urging a reversal of the university decision.

The Grutter ruling “swung open the doors to regimes that discriminate on the basis of race and ethnicity,” Mr. Bush, a Republican, said.

The punitive-damages dispute, which concerns a $79.5 million award to the widow of a former smoker, also may hinge on the approach of Justices Roberts and Alito to precedent. Three justices — Antonin Scalia, Clarence Thomas, and Ruth Bader Ginsburg — have never accepted the court’s decade-old ruling that imposed the first constitutional limits on awards.

How much respect Justices Roberts and Alito will accord to precedent isn’t yet clear. At their Senate hearings, both nominees left themselves flexibility, pointing to the landmark 1954 Brown v. Board of Education school desegregation case as an instance in which the court properly abandoned a precedent.

One of the few hints about their views came in a campaign finance case resolved by the court in June. Justice Breyer’s lead opinion invoked the stare decisis principle in refusing to let a state impose spending limits on candidates.

Justice Alito opted not to join that section of Justice Breyer’s opinion, saying the issue wasn’t directly presented in the case. Justice Roberts, however, joined the passage, which said, “Departure from precedent is exceptional and requires special justification.”

Some court observers question whether Justice Roberts would go along with a decision to overturn either the abortion or the affirmative action precedent.

“Given his institutional role as chief and given his new tenure on the court, I don’t think he wants to send that kind of message of instability,” the ACLU lawyer, Mr. Shapiro, said. “I don’t think that without his support either of those two cases are really in play.”