President Reagan and Prime Minister Margaret Thatcher talk in New York in 1985. (MIKE SARGENT/AFP/Getty Images)

U.K. Prime Minister Margaret Thatcher, who died Monday, and U.S. President Ronald Reagan are remembered as a geopolitical "power couple," a partnership that pushed for free-market conservatism and helped win the Cold War. In both U.S. and U.K. politics, their names are practically synonymous.

But the truth was far more complicated and, particularly when it came to the more difficult moments of the Cold War, Reagan and Thatcher found plenty to disagree on. Nicholas Henderson, the U.K. ambassador to Washington under Thatcher, was later asked by a British politician if he had learned any real secrets. He paused before saying, "If I reported to you what Mrs. Thatcher really thought about President Reagan, it would damage Anglo-American relations."

That quote was revealed in a book released last year by historian Richard Aldous, "Reagan and Thatcher: The Difficult Relationship". The book argues that the two leaders clashed repeatedly and that their partnership was in some ways stage-managed, a show of unity that covered over the disagreements over sometimes major points. Partly it was for British and American audiences, who respected the other nation's leader and enjoyed seeing them joined at the hip. And partly it was for Moscow and for Soviet bloc nations, a message that at a time of geopolitical upheaval – and shows of fracture in Eastern Europe – there would be no space between the West's two strongest nations and opponents of the Soviet Union.

That show of unity worked, finally convincing the world that Joseph Stalin had been utterly wrong to predict the break-up of the Western nations, but it was so effective that it has persisted, obscuring some of the more complicated truths of their relationship.

Many of those disagreements focused on the one area where Reagan and Thatcher supposedly stood in perfect harmony: the Cold War and the Soviet Union. She pushed early for the Western world to work with an upstart reformer in Soviet politics named Mikhail Gorbachev, although the Reagan administration was initially far more skeptical. (One of the harshest skeptics of Gorbachev was Robert M. Gates, who later became defense secretary under George W. Bush and Barack Obama.) Even after she left office in 1990, Thatcher continued to push Washington, urging then-president George H.W. Bush to show him public support.

There were other disagreements. Nuclear weapons were a particularly fraught issue: Reagan worked to reduce the number of warheads in the world. Thatcher later said, according to David Hoffman's Pulitzer-winning history of the arms race, that she was "horrified" by his efforts and his plan to reach a global zero. After a meeting with Gorbachev in which he warned of "nuclear winter," a phrase that Reagan would later take up himself, Thatcher sighed, "I was not much moved by all this."

In his second term, Reagan held a series of nuclear negotiations with Soviet Premier Mikhail Gorbachev, in which both heads of state openly shared, and at points almost reached agreement on, a goal of reducing the world's warhead count to zero. Thatcher publicly condemned nuclear disarmament. And though careful not to mention Reagan or his goals, the message was surely not lost on Washington. "A world without nuclear weapons may be a dream but you cannot base a sure defence on dreams," she said in 1987. "Without far greater trust and confidence between East and West than exists at present, a world without nuclear weapons would be less stable and more dangerous for all of us."

In 1982, the Reagan administration showed tepid and reluctant support for the U.K.'s decision to send a fleet to protect its Falkland Islands territory from Argentina. Though the United States joined the United Kingdom in vetoing a United Nations Security Council resolution calling for an immediate ceasefire on the Falklands, the U.S. ambassador to the U.N. Jeane Kirkpatrick quickly said the vote had been a mistake, humiliating Thatcher. The night of the invasion, Kirkpatrick dined at the Argentine ambassador's house.

The next year, when the United States invaded Grenada, a former U.K. colony in the Caribbean, Thatcher told the Irish premier, according to Aldous, "The Americans are worse than the Soviets ... persuading the governor [of Grenada] to issue a retrospective invitation to invade after they had taken him aboard an American warship." The same year, when Reagan withdrew troops from Lebanon after a bombing, Thatcher called his move "a naïve failure" that was part of his administration's "amateurishness and disarray" in the Middle East.

Still, they agreed on major points: Reagan's plan to end detente with the Soviet Union and push for global communism's collapse, close security cooperation and an emphasis on economic and social conservatism. And they agreed that it was important to show a united front to Moscow, which had been looking for decades for signs of a split between the Western powers. But the many disputes along the way were not always minor, particularly on nuclear weapons. Reagan and Thatcher were indeed close allies, but they still led different nations with different goals and different strategies.

As the English statesmen Henry Temple said in 1848 of his nation's foreign policy, "We have no eternal allies, and we have no perpetual enemies. Our interests are eternal and perpetual, and those interests it is our duty to follow."

You might recognize that quote in its incorrect but more commonly attributed version, "England has no friends, only interests."