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The extraordinary thing about this handshake was that it happened in space, somewhere over West Germany, in a joint mission that symbolized a thawing of tensions in the long-running Cold War.
"We estimate over a billion to a billion and a half people around the world saw me shake hands with Alexey and said look if people like this can work together we can work together on a lot of things," the 84-year-old Stafford said.
Up until then, space had been the stage for one of the most riveting rivalries between Russia and the United States as both nations competed for superiority in the wake of World War II. It was known as the space race
"I think that these were the best competitions, higher than Olympics," the 81-year-old Leonov said, "because as a result of this, as you call it race, we developed brand new equipment that we're still using."
Leonov had played a key part in the contest which began with the launch of the artificial satellite Sputnik into orbit. This world-first event was a worrying sign for America at a time of increased tensions. It implied that Russia also had the capacity to potentially launch a long-range nuclear missile.
Russia continued to clock up the milestones, sending the first animal, man and then woman into space. When Leonov trumped U.S. plans to pioneer space walking in March 1965, it appeared that Russia had the race won.
But four years later, America would claim the ultimate prize, when Neil Armstrong and Buzz Aldrin landed on the Moon
Both nations recognized the strengths and weaknesses in each other's space programs, so when political tensions eased following the end of the Vietnam War, plans were made for a joint mission known as the Apollo-Soyuz Test Project.
The primary goal was to design and build an international docking system that would allow two different spacecrafts to join in orbit, allowing crews to cooperate on experiments and further exploration.
After four years of planning, on July 15, 1975, Russian cosmonauts Alexey Leonov and Valeri Kubasov were launched into space aboard a Soyuz spacecraft from the Baikonur Cosmodrome in the Kazakh desert.
Seven and a half hours later, the American spacecraft Apollo was sent on its last ever mission, guided by astronauts Donald Slayton, Vance Brand and Thomas Stafford.
Two days later, the Russians and Americans met in space and made history.
At the time, Stafford said: "When we opened this hatch in space, we were opening, back on Earth a new era in the history of man."
But the period of detente that enabled the Apollo-Soyuz Test Project to proceed, ended when Russia and the U.S. found themselves in a proxy war in Afghanistan at the end of the decade.
It would be another 20 years before Russia and the U.S. docked in space again, with the MIR- Shuttle Program.
Another two decades on, Russia's annexation of Crimea has seen tensions rise again
, but friendly cooperation continues in orbit aboard the International Space Station.
"We rely on each other literally for our lives," American astronaut Mark Kelly said from the International Space Station, "so despite any political differences our countries may have, or past history, we get along just great."
So too, Leonov and Stafford. The two men remain close friends and their affection for each other was clear as they met at Moscow's Space Museum on Wednesday to celebrate the 40th anniversary of the launch of the Apollo-Soyuz Test Project.
Even when they re- enact the famous handshake for the cameras, it is with vigor and for them remains symbolic.
Leonov even borrows from a speech his American counterpart made to students at the University of Moscow when they returned from their pioneering mission 40 years ago: "Cooperation means friendship, friendship means peace," he quotes. "Let's live in peace."