“Police officers must serve and protect people on both sides of any given issue,” Mr. Guglielmi said. “They can’t effectively do that if they are perceived to take one side over another. So when officers are on duty and wearing the uniform, there are no sides, just service to anyone who needs them.”

Asked whether that meant the officers who posed with Ms. Clark would not have been reprimanded if they had taken the photo while off duty and out of uniform, he said, “Absolutely.”

The photo was particularly resonant in Chicago, a city where the Justice Department found in January that the police had frequently violated civil rights; where civil rights groups filed a lawsuit in June alleging police abuses; where protests erupted over the police shooting of 17-year-old Laquan McDonald; where an officer was convicted in August for shooting two black teenagers in a car that was driving away from him. The city’s police and politicians have been trying in various ways to gain residents’ trust.

Mayor Rahm Emanuel told reporters on Tuesday that he supported the decision to reprimand the officers, and that the Police Department’s policy was consistent. But he acknowledged that the officers “were somewhat betwixt, between two different aspirations”: the desire to follow department rules and the desire to improve relations between the police and the public, The Sun-Times reported.

Kneeling took on popular significance as a form of protest in 2016, when the National Football League player Colin Kaepernick did it during the national anthem. But when President Trump denounced it last week, what had been isolated actions exploded into a national display of defiance, with players from New York to Indianapolis to Los Angeles dropping to their knees on the sidelines of their Sunday games.

In their photo, Ms. Clark and the two officers paired kneeling with another symbol of protest, made famous by the American runners John Carlos and Tommie Smith on the medal podium at the 1968 Summer Olympics: raised fists and, from Ms. Clark, a downward gaze.

“I’m not going to say anything that is going to get those officers in trouble,” Ms. Clark told a television interviewer when asked whether the officers had expressed any awareness that what they were doing was against the rules. “I think that picture speaks volumes, and I think it was enough.”