By BRENT KENDALL

South Carolina Attorney General Alan Wilson, an opponent of the health‐care law, would like to think Justice Anthony Kennedy is on his side, citing the moment Tuesday when “he said the health‐care mandate changes the relationship of the federal government to the individual in a very fundamental way.”

Doug Kendall, president of the Constitutional Accountability Center, a liberal legal group that filed briefs supporting the overhaul, believes Justice Kennedy might be on his side, based on his description of the problems created by the uninsured.

Steven A. Engel, a former law clerk of Justice Kennedy and now a partner at Dechert LLP said: “It’s entirely possible he doesn’t know yet which way he’s going to go.”

The mystique that is Justice Kennedy—the almost‐certain swing vote in any big Supreme Court case— was on full display during Tuesday’s arguments over whether President Barack Obama’s signature legislative achievement passes constitutional muster. After two intense hours of oral argument, both sides were investing their hopes in him. But neither could confidently predict how he will vote.

George Washington University law professor Orin Kerr, a former Justice Kennedy law clerk, said his former boss appeared to be struggling with how the case should be decided. “It’s too hard to predict where he will come down,” Mr. Kerr said. “Some justices have a clear view right off the bat. Justice Kennedy is more likely to think over his position over time.”

The true leanings of Justice Kennedy, an appointee of President Ronald Reagan who has served since 1988, can be tough to discern from his comments during oral arguments.

Take his October 2004 questions in Roper v. Simmons, where the court considered whether the Constitution forbids the death penalty for juvenile offenders. During that session, he asked several skeptical questions and suggested the death penalty served as a deterrent to 16‐ and 17‐year‐old gang members. His comments left some court watchers believing he would side with death penalty supporters.

Five months later, Justice Kennedy wrote the court’s 5‐4 opinion declaring juvenile executions a violation of the Constitution’s ban on cruel and unusual punishment. Four liberal members joined him in the majority.

Being the swing vote, Justice Kennedy has often found himself at the center of the court’s biggest cases. In 2003, he wrote a passionate majority opinion in Lawrence v. Texas, where the court voted 6‐3 to strike down a Texas anti‐sodomy statute.

More recently, he wrote the court’s 2010 divided ruling in Citizens United v. Federal Election Commission, which struck down long‐standing limits on corporate election expenditures.

Before Tuesday, the White House expected an easier reception. Justice Kennedy proved harsher than government lawyers had predicted, said an administration official who attended the argument. But the official noted that while other justices often reveal their leanings at oral arguments, Justice Kennedy’s questions tend only to indicate the issues he finds most vexing. And in this case, the official said, Justice Kennedy seemed as aware of the peculiar problems afflicting the health‐care market as he was troubled by the novelty of the method Congress adopted to cure them.

— Jess Bravin contributed to this article.