News of a “Stop the Steal” symbol that flew at the home of Justice Samuel A. Alito Jr. after the 2020 election has elicited concerns from politicians, legal scholars and others about the Supreme Court’s ethical standards — and, most urgent, whether the public will regard its rulings about Jan. 6, 2021, as fairly decided.

In coming weeks, the Supreme Court is expected to issue two key decisions involving the storming of the Capitol on that day. The cases will shape the degree to which former President Donald J. Trump can be held accountable for his efforts to subvert the election.

“These cases were always going to be seen through an ideological and partisan lens,” Michael C. Dorf, a Cornell law professor and former clerk to Justice Anthony Kennedy, said in an interview. “The fact that you’ve got two justices who are strongly identified not just with the Republican Party or the conservative movement but the ‘Stop the Steal’ movement raises that to another level.” Justice Clarence Thomas’s wife, Virginia Thomas, was directly involved in efforts to overturn the election.

An upside-down flag, a popular symbol with Trump supporters contesting President Biden’s victory, appeared on Justice Alito’s front lawn in January 2021, The New York Times reported based on photographs and interviews with neighbors. It hung on the Alitos’ flagpole days before the inauguration, a little over a week after the Capitol riot and while the Supreme Court was considering taking up an election case.

Justice Alito acknowledged the flag in a statement to The Times, saying that it was part of a dispute that his wife, Martha-Ann Alito, had with neighbors and that he had “no involvement whatsoever.”

In an interview with Fox News on Friday, Justice Alito said his wife put up the flag after a neighbor down the street posted an anti-Trump sign with an expletive near a school-bus stop. After his wife spoke with the neighbors about the sign, the neighbors put up another that blamed her for Jan. 6, the justice told Fox. Following an argument with those neighbors during a stroll, Mrs. Alito put up the flag in response, he said.