Several officers interviewed in four cities on Friday said they tried to ignore such remarks. Others said they had zero tolerance for being treated disrespectfully in public.

The line of when to put on handcuffs is a personal and blurry one, varying among officers in the same city, the same precinct, even the same patrol car.

A mounted police officer who has been with the Los Angeles Police Department for 25 years said that taking verbal abuse was a regular part of his job.

“We don’t get to tell people what they want to hear,” said the Los Angeles officer, who, like others interviewed for this article, spoke on the condition of anonymity to avoid being quoted on duty. “Whether we’re giving them a ticket or responding to some conflict between a husband and wife, we’re not dealing with people at their best, and if you don’t have a tough skin, then you shouldn’t be a cop.”

The officer said he recently confronted a woman walking in the middle of the street and asked her to step out of traffic. She refused and became belligerent, using a string of four-letter words and ethnic epithets. He said he wrote the woman a ticket and went on his way.

But in Brooklyn, a 24-year-old officer, with three years on the force, seemed less inclined to walk away from verbal abuse.

“We say, ‘Back down,’ ” he said. “If they don’t back down and start making direct threats, that’s an offense. They don’t get a free pass.”

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He said that threats could be defined in different ways, and he preferred to talk people down, but that the rules changed if a crowd formed, which was routine in New York and also occurred during the Gates incident.

“I wouldn’t back down if there’s a crowd gathering,” the Brooklyn officer said, in part out of concern of sending a message of weakness that could haunt another officer later. “We’re a band of brothers. We have to be there to help each other out. If there’s a group and they’re throwing out slurs and stuff, you have to handle it.”

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A 13-year veteran of the Denver police force, who did not wish to give his name, said likewise. “We’re not going to take abuse,” he said. “We have to remain in control. We’re running the show.”

But Robert Anderson, with the same department five years, said he tried to “let people vent” if they grew irate. “People usually aren’t happy to see the police,” he said. “They’d rather see a fireman.”

In New York, State Senator Eric Adams, a retired New York City police captain and co-founder of the group 100 Blacks in Law Enforcement, said the rules for dealing with someone differed by setting.

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“If it’s their house, they’re allowed to call you all sorts of names,” Mr. Adams said. “A man’s house is his castle. If they’re in the street, and they don’t listen to the officer’s warning, ‘Sir, you’re being disorderly,’ you can lock them up at this time.”

Not that the officer necessarily should, he said.

“Let’s say I do a stop,” Mr. Adams said. “I question, and it’s nothing. ‘Sir, I’m sorry, I apologize.’ What’s the reason for staying, if the anger’s directed at me? If it’s directed at a third party, a storekeeper, I stay.”

But if the officer himself is the provocation, the officer should leave, he said, and added that Sergeant Crowley did not use such common sense.

Michael J. Palladino, president of the Detectives Endowment Association in New York, took a harder line and said officers should not tolerate disrespect on the street.

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“We pay these officers to risk their lives every day,” Mr. Palladino said. “We’re taught that officers should have a thicker skin and be a little immune to some comments. But not to the point where you are abused in public. You don’t get paid to be publicly abused. There are laws that protect against that.”

In Atlanta, Officer M. Tate, who would not give his first name, said he was trained not to lose his cool — or his job — by reacting to name calling. He recalled from memory the exact definition of when a person’s behavior crossed the line into being worthy of arrest: “The set of circumstances that will lead a reasonable and prudent person to believe that a crime has or is about to be committed and that the person in question is involved in a significant manner.” Anything short of that, he said, does not warrant handcuffs.

“I’ll take them yelling at me,” Officer Tate said. “Unless I’m hit or they get violent, I won’t arrest them for just yelling at me.”

But the training cannot be applied to every situation, one officer said.

“You want the training?” a detective in Queens asked. “Or how I train myself?”

He described a scenario he had faced many times: stopping someone who he just saw appear to slip drugs to someone else, only to learn that was not the case. “ ‘Oh, it’s a cigarette. Oh, O.K., sorry to bother you,’ ” the detective said.

And if the person then becomes verbally abusive?

“If you locked everybody up that was technically disorderly — you’ve got to know which battles to fight,” he said. “If this guy’s causing commotion, there’s a scene, you look for the level-headed person who’s a friend of his. Say, ‘Look, we’re out here cleaning up your block.’ When you leave, they’re going to talk to him.”

Senator Adams said black men were more likely to be locked up for what in police parlance is called getting “lippy.”

“The ‘uppity Negro,’ ” he said. “You may not have committed a crime, but you know what? You’ve got a big mouth.”