It’s one of the iconic images of the Vietnam War. Hubert van Es’ snapshot shows a military helicopter perched atop the US Embassy in Saigon as Americans stream up a stairway, desperately boarding the last flight out.

Except that isn’t a military helicopter, that’s not the US Embassy, those aren’t Americans and that wasn’t the last flight out.

And, perhaps most surprising of all, the photo is not about defeat.

“As I was researching it, I found that almost everything I had assumed about that photo was wrong,” historian and author Thurston Clarke tells The Post.

The surprising tale of that photo, along with a complete story of the last days of the Vietnam War, is told in Clarke’s new book, “Honorable Exit: How a Few Brave Americans Risked All to Save Our Vietnamese Allies at the End of the War” (Doubleday).

It’s a nearly minute-by-minute account of the United States’ pullout from the quagmire conflict, and much of the story is encapsulated in that single photo, taken on April 29, 1975.

By early 1975, America’s prospects in Vietnam were looking bleak. The communists continued to push south, capturing territory and sending thousands of refugees fleeing for safety.

‘As I was researching it, I found that almost everything I had assumed about that photo was wrong.’

After Da Nang fell, World Airlines, a government contractor tasked with flying troops and equipment in and out of the country, attempted that March to evacuate people from the city, leading to a chaotic scene as panicked Vietnamese tried to push their way on.

The plane, under heavy fire, managed to get airborne, but stowaways and those clinging to the side plummeted to their deaths, leaving corpses littering the runway.

When President Gerald Ford saw footage of the fiasco, he remarked to a friend, “That’s it. It’s time to pull the plug. Vietnam is gone.”

“The news reports, photographs, and films of Da Nang … received wide coverage in the United States,” Clarke writes. “Instead of unleashing Americans’ charitable impulses, the news convinced them that the United States should avoid becoming reinvolved in the war, even on a humanitarian basis, and opinion polls showed a majority of Americans opposed to accepting Vietnamese refugees.”

Yet American leaders took a different path. They quickly began planning to evacuate as many Vietnamese as possible and resettle them in the US.

Secretary of State Henry Kissinger argued at an April 2 State Department staff meeting that it was America’s duty “to get the people who believed in us out.”

A list of potential Vietnamese evacuees was compiled. It included high-ranking officials, translators and others at high risk.

Kissinger wired the US Vietnamese ambassador, Graham Martin, that failure to address the “difficult question of evacuation” could result in the loss of American “lives, national dignity, and a common sense of confidence that we can manage whatever crisis the future may hold.”

By April 27, 1975, North Vietnamese troops had surrounded Saigon and it became clear an attack was imminent. Chaos erupted as residents scrambled to flee. They tried to push their way into the US Embassy in hopes of escaping.

“Women outside the walls wailed and wept,” the author writes. “Some rolled on the sidewalk in hysterics, crying out the names of the American servicemen, businessmen and diplomats whom they had served as clerks, drivers, cooks, cleaners, bodyguards and interpreters. Mothers tried to pass their babies to Marines.”

The US finally initiated an operation known as Frequent Wind, which would helicopter evacuees out of Saigon to US ships offshore for eventual transport to America.

In early April, the Americans had scouted rooftops as potential helicopter landing sites, choosing 13 across Saigon. US contractors arrived to remove flagpoles and laundry lines, and built whatever infrastructure was needed.

On April 29, copters fanned out across the city, picking up the hopeful. The Hueys were operated by Air America, a line covertly run by the CIA. The 31 pilots flying the hazardous missions were volunteers. Nearly everyone who was asked stepped forward.

Besides the embassy, one of the pickup points was the Lee Hotel at 6 Chien Si Circle. There, the ambassador had assembled a group of VIPs for evacuation.

Manning one of the copters was O.B. Harnage, a macho, cigar-smoking, hard-drinking CIA officer who wore an eye patch after he was wounded by shrapnel in World War II.

Harnage’s helicopter landed at the Lee Hotel, and the aircraft was quickly mobbed by desperate Vietnamese. One South Vietnamese soldier pushed toward Harnage and pulled the pin from a grenade, telling him, “We go with you or I drop the grenade.”

Harnage shoved the barrel of a machine gun in the man’s face and responded, “Go ahead. You’ll never hear it go off.” The soldier backed off.

After the Lee Hotel became unsafe, the CIA station chief urged Harnage to start picking up at 22 Gia Long Street. The apartment building’s top floor served as quarters for CIA employees.

VIPs and other Vietnamese had been gathering at the location in hopes of a rescue. Wooden stairs had been built to the top of the roof’s elevator shaft to aid the evacuation.

Harnage made three pickups, packing 20 people onto each flight. To make room for them all, he rode outside the bird, standing on the skid while gripping a machine gun.

“Remember, he wasn’t taking people he worked with or he knew,” the author says. “They were perfect strangers and he was risking his life.”

The station chief had ordered him to concentrate on the VIPs, but Harnage decided to board on a first-come, first-serve basis. Some parents handed him their children with heartbreaking notes pinned to the kids’ clothes, such as, “My son wants to be a doctor” and “My daughter is very musical.”

When a burly Korean diplomat tried to elbow other evacuees aside to make it onto the chopper, Harnage punched the man, leaving him bloody and chastened. (He was allowed on a later flight.)

Harnage refused to let anyone board with more than one small bag, tossing everything else over the side of the building — including what was likely millions of dollars of hoarded gold.

On the fourth touchdown at Gia Long Street, van Es was sitting in the offices of United Press International about four blocks away when a colleague called out, “Van Es, get out here! There’s a chopper on that roof!”

Van Es grew up in the Netherlands and decided to become a war photographer after seeing Frank Capra pictures. He first worked in Hong Kong for the South China Morning Post before heading to Vietnam to work for NBC, then UPI.

He grabbed his Nikon and a 300mm lens and began snapping.

The shot that went around the world showed Harnage, dressed in a white shirt and clutching a cigar between his teeth, reaching down the stairs to grab Thiet-Tan Nguyen, a young doctor who would ultimately become an anesthesiologist in California.

Next is Tong Huyhn, another doctor who would eventually settle near Atlanta. The slender teenage girl is Tuyet-Dong Bui, who would go on to earn a degree in microbiology before becoming a biotech researcher.

Just below her is her brother, who had traded his motorbike to the driver of a high-ranking military official in exchange for being shown to the building.

Also on the ladder is South Vietnamese Minister of Defense Tran Van Don.

The photo marked the last trip Harnage was able to make to the location, leaving dozens of people behind, watching the skies for a helicopter that would never appear.

Flights continued elsewhere through the night. In the end, the airlift extracted some 1,373 Americans and 5,595 Vietnamese from Saigon in less than 24 hours.

Harnage returned to America and became a real estate salesman. He died in 2008. Van Es was paid a one-time bonus of $150 for the photo. He continued covering conflicts, and died in 2009.

Clarke says he hopes van Es’ famous photo will be viewed in a new light.

“At the time, this photo seemed to be a metaphor for Vietnam — our first lost war and us scrambling away from the roof of the embassy,” the author says. “It was a damning photo that fit into the narrative of the time.

“It was anti-heroic, but when you look at what’s actually going on on that rooftop, this was a moment of considerable heroism.”