The city’s infamous rubber rooms have rebounded.

In one of the “reassignment centers,” 16 exiled educators sit in a city Department of Education building in Long Island City, Queens, including a dozen packed into one room — where they do virtually no work.

They listen to music, do crossword puzzles, chat — and as this exclusive Post photo reveals, doze on the taxpayer’s dime.

The rules forbid beach chairs and air mattresses, but not nap time. The teacher sprawled on the floor, pulled a wool hat over his eyes to shut out the fluorescent lights and slept.

Others prop up two chairs to recline or just lay their heads on the table. “It’s gone right back to the way it was in the old days, an old-fashioned rubber room,” one banished teacher said.

Despite the photographic evidence and teacher testimony to the contrary, the city denies the existence of the derided holding pens. “There are no more rubber rooms,” DOE officials told The Post last week, saying reassigned staffers are given “administrative duties.”

In 2010, the DOE and the teachers union trumpeted a major agreement to close the centers holding more than 700 idled educators accused of misconduct or incompetence.

Many teachers settled charges by paying fines and finally returned to classrooms, while those still expelled were scattered across the five boroughs. But the rubber room deal is routinely violated. “No one pays any attention to the agreement,” said Betsy Combier, a veteran paralegal who helps defend teachers.

The DOE refused to say how many removed teachers and other tenured staffers remain in limbo, but sources estimate 200 to 400 get paid while awaiting disciplinary hearings. Their salaries total $15 million to $20 million a year.

The Long Island City castoffs begin their day by reading newspapers, then turn on the radio. They get 45 minutes to leave for lunch. They chat and sometimes exercise to “relieve the stress.”

While the city promised to keep removed educators busy, the Queens exiles say they only occasionally oblige requests to do menial tasks like stuffing folders or making copies. Others refuse to do such work, calling it “demeaning.”

They mainly just kill time to get through a six-hour, 20-minute day.

“I’m so exhausted from being in this place doing nothing,” one said.

Several teachers on the payroll have been benched for up to five years due to a stunning bureaucratic breakdown. The 2010 deal required the independent arbitrators who conduct termination trials to issue a decision 30 days after a hearing, so that vindicated teachers could return to work and bad ones could be axed.

But decisions still come months — or even years — late. The DOE says it can’t enforce the rule.

“They’re just letting me sit here,” said a teacher removed from the classroom nearly five years ago on charges of physically abusing children, which he denies. His trial ended four years ago. He makes about $70,000 a year.

Social Studies teacher David Suker has sat in the rubber room since September 2015, even though a state Supreme Court judge overturned his termination. Suker, an Army vet who taught at-risk kids in The Bronx, was found guilty of failing to immediately report his arrest in the Occupy Wall Street protests but was finally fined just $7,000.

After The Post asked why the DOE had not put Suker back to work, officials on Friday assigned him to a school as a substitute.