(CNN) It's a detail that stands out in photos of recent immigration roundups across the country: federal agents wearing clothing that labels them as "police."

Immigration and Customs Enforcement officials say it's an accurate word for them to use, and it can mean the difference between life and death in volatile situations.

But elected officials in Los Angeles say they want the practice to stop. When ICE officers call themselves "police," they claim, it's misleading and endangers public safety.

How long has this been happening? What's the debate? And why is it coming up now?

The history

It isn't new for ICE officers and agents to refer to themselves as police.

Photos of operations dating back years show them wearing jackets and vests with that label, usually in large letters that read, "POLICE ICE."

An ICE agent in San Diego directs a group of men being deported to Mexico in February 2015.

Sarah Rodriguez, an ICE spokeswoman, says ICE agents may sometimes verbally identify themselves as police when they're going into a situation.

"It is the universally recognized term for law enforcement and our personnel routinely interact with individuals from around the world," she said. "In the often dangerous law enforcement arena, being able to immediately identify yourself as law enforcement may be a life-or-death issue."

The debate

This approach has drawn renewed criticism from activists and some officials amid heightened publicity about immigration crackdowns since President Donald Trump took office.

Several Los Angeles leaders are asking immigration officials to put a stop to the practice, calling it "corrosive." They argue it undercuts years of police efforts to build trust in immigrant communities and will make witnesses and victims less likely to come forward when crimes are committed.

"Especially in these turbulent and uncertain times, we urge that ICE agents operating in Los Angeles immediately stop representing that they are 'police' officers," the city's mayor, attorney and City Council president wrote in a letter last month.

The elected officials said they sent the letter after seeing a Los Angeles Times report about the practice. That article's headline: "It's legal for an immigration agent to pretend to be a police officer outside someone's door. But should it be?"

Rodriguez, the ICE spokeswoman, says the term "police" is accurate and there's nothing misleading about using it.

"It's clear that we are a law enforcement agency," she said. "We have police authority."

ICE agents detain a man in Los Angeles in October 2015.

A Los Angeles police spokesman declined to comment. Los Angeles City Attorney Mike Feuer told CNN this week that the issue remains a significant concern.

"Irrespective of whether it's lawful to do that, that begs the question of whether it's ethical to do it or whether it's appropriate policy to do it. It begs the question of whether ICE doing so endangers public safety, which it does," he said. "ICE misidentifying itself as police officers in my city makes Los Angeles less safe for everyone."

He pointed to a recent example that drew widespread attention: a video of ICE agents -- wearing jackets that say only "police" -- arresting a father who was dropping off his kids at a Los Angeles school.

"That, too, sends a chilling message to our immigrant communities," he said.

Led by US The presence of the word "police" on ICE jackets and vests also has drawn criticism from lawmakers.Led by US Rep. Mike Thompson , D- California, dozens of Democratic congressmen signed a letter last month asking Homeland Security Secretary John Kelly to take action.

"Local law enforcement in our Congressional districts have expressed serious concerns that this practice causes confusion and undermines their officers' efforts to build trust in our immigrant communities," the letter said. "We respectfully urge you to direct ICE to remove the word 'Police' from all ICE gear."

The bigger picture

This dispute is just one of the ways some local and federal authorities are sparring over immigration.

The larger battlefront on the horizon: debate over so-called sanctuary cities.

It's a broad term describing local governments that have enacted policies limiting their cooperation with federal immigration enforcement. Their goal: to protect undocumented immigrants who are not otherwise engaged in criminal activity from being detained or deported.

In Los Angeles, for example, police have a policy of not stopping people solely to find out their immigration status. It's been in place since 1979.

In an interview last month with CNN affiliate KABC , Police Chief Charlie Beck said he wanted to reassure the city's undocumented residents that the policy isn't changing.

"Your status has nothing to do with your contact with Los Angeles Police Department," Beck said. "We will not give you to ICE. We will not refer you to ICE."

Those who support sanctuary policies say they build trust with local law enforcement and keep communities safer. ICE has said such policies inhibit their ability to enforce laws.

This much is clear: The fight's not over.

Trump signed an executive order in January pledging to block sanctuary jurisdictions from receiving federal grant money. Leaders of sanctuary cities across the country have vowed not to back down.

Legal experts say it's likely the two sides will face off in court.