WASHINGTON—U.S. President Donald Trump was already revved up when he emerged from his limousine to visit NATO’s new headquarters in Brussels in May. He had just met France’s recently elected president, Emmanuel Macron, whom he greeted with a white-knuckle handshake and a complaint that Europeans do not pay their fair share of the alliance’s costs.

On the long walk through the NATO building’s cathedrallike atrium, the president’s anger grew. He looked at the polished floors and shimmering glass walls with a property developer’s eye. (“It’s all glass,’’ he said later. “One bomb could take it out.”) By the time he reached an outdoor plaza where he was to speak to the other NATO leaders, Trump was fuming, according to two aides who were with him that day.

He was there to dedicate the building, but instead he took a shot at it.

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“I never asked once what the new NATO headquarters cost,” Trump told the leaders, his voice thick with sarcasm. “I refuse to do that. But it is beautiful.” His visceral reaction to the $1.2-billion (U.S.) building, more than anything else, coloured his first encounter with the alliance, aides said.

Nearly a year into his presidency, Trump remains an erratic, idiosyncratic leader on the global stage, an insurgent who attacks allies the United States has nurtured since the Second World War and who can seem more at home with America’s adversaries. His Twitter posts, delivered without warning or consultation, often make a mockery of his administration’s policies and subvert the messages his emissaries are trying to deliver abroad.

Trump has pulled out of trade and climate change agreements and denounced the 2015 nuclear deal with Iran. He has broken with decades of U.S. policy in the Middle East by recognizing Jerusalem as the capital of Israel. And he has taunted Kim Jong Un of North Korea as “short and fat,” fanning fears of war on the peninsula.

He has assiduously cultivated President Xi Jinping of China and avoided criticizing President Vladimir Putin of Russia — leaders of the two countries that his own national security strategy calls the greatest geopolitical threats to America.

Above all, Trump has transformed the world’s view of the United States from a reliable anchor of the liberal, rules-based international order into something more inward-looking and unpredictable. That is a seminal change from the role the nation has played for 70 years, under presidents from both parties, and it has lasting implications for how other nations chart their futures.

Trump’s unorthodox approach “has moved a lot of us out of our comfort zone, me included,” the national security adviser, Lt. Gen. H.R. McMaster, said in an interview. A three-star Army general who served in Iraq and Afghanistan and wrote a well-regarded book about the White House’s strategic failure in Vietnam, McMaster defined Trump’s foreign policy as “pragmatic realism” rather than isolationism.

“The consensus view has been that engagement overseas is an unmitigated good, regardless of the circumstances,” McMaster said. “But there are problems that are maybe both intractable and of marginal interest to the American people, that do not justify investments of blood and treasure.”

Trump’s advisers argue that he has blown the cobwebs off decades of foreign policy doctrine and, as he approaches his anniversary, that he has learned the realities of the world in which the U.S. must operate.

They point to gains in the Middle East, where Crown Prince Mohammed bin Salman is transforming Saudi Arabia; in Asia, where China is doing more to pressure a nuclear-armed North Korea; and even in Europe, where Trump’s criticism has prodded NATO members to ante up more for their defence.

The president takes credit for eradicating the caliphate built by Daesh, also known as ISIS or ISIL, though he mainly accelerated a battle plan developed by his predecessor, Barack Obama. His aides say he has reversed Obama’s passive approach to Iran, in part by disavowing the nuclear deal.

While Trump has held more than 130 meetings and phone calls with foreign leaders since taking office, he has left the rest of the world still puzzling over how to handle an American president unlike any other. Foreign leaders have tested a variety of techniques to deal with him, from shameless pandering to keeping a studied distance.

“Most foreign leaders are still trying to get a handle on him,” said Richard N. Haass, a top State Department official in the George W. Bush administration who is now the president of the Council on Foreign Relations. “Everywhere I go, I’m still getting asked, ‘Help us understand this president, help us navigate this situation.’

“We’re beginning to see countries take matters into their own hands,” Haass continued. “They’re hedging against America’s unreliability.”

Difficulties with Merkel

Few countries have struggled more to adapt to Trump than Germany, and few leaders seem less personally in sync with him than its leader, Chancellor Angela Merkel, the physicist-turned-politician. After she won a fourth term, their relationship took on weighty symbolism: The great disrupter versus the last defender of the liberal world order.

In one of their first phone calls, the chancellor explained to the president why Ukraine was a vital part of the trans-Atlantic relationship. Trump, officials recalled, had little idea of Ukraine’s importance, its history of being bullied by Russia or what the U.S. and its allies had done to try to push back Putin.

For Merkel and many other Germans, something elemental has changed across the Atlantic. “We Europeans must really take our destiny into our own hands,” she said in May. “The times in which we can fully count on others — they are somewhat over.”

Better relations with autocrats

There have been fewer misunderstandings with autocrats. Xi of China and King Salman of Saudi Arabia both won over Trump by giving him a lavish welcome when he visited. The Saudi monarch projected his image on the side of a hotel; Xi reopened a long-dormant theatre inside the Forbidden City to present him and his wife an evening of Chinese opera.

“Did you see the show?” Trump asked reporters on Air Force One after he left Beijing in November. “They say in the history of people coming to China, there’s been nothing like that. And I believe it.”

Later, chatting with his aides, Trump continued to marvel at the respect Xi had shown him. It was a show of respect for the American people, not just for the president, one adviser replied gently.

Then, of course, there is the strange case of Putin. The president spoke of his warm telephone calls with the Russian president, even as he introduced a national security strategy that acknowledged Russia’s efforts to weaken democracies by meddling in their elections.

Trump has had a bumpier time with friends. He told off Prime Minister Theresa May on Twitter, after she objected to his exploitation of anti-Muslim propaganda from a far-right group in Britain.

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“Statecraft has been singularly absent from the treatment of some of his allies, particularly the U.K.,” said Peter Westmacott, a former British ambassador to the U.S.

Trump’s feuds with May and other British officials have left him in a strange position: feted in Beijing and Riyadh but barely welcome in London, which Trump is expected to visit early next year, despite warnings that he will face angry protesters.

Aides to Trump argue that his outreach to autocrats has been vindicated. When the Saudi crown prince visited the White House in March, the president lavished attention on him. Since then, they say, Saudi Arabia has reopened cinemas and allowed women to drive.

But critics say Trump gives more than he gets. By backing the 32-year-old crown prince so wholeheartedly, the president cemented his status as heir to the House of Saud. The crown prince has since jailed his rivals as Saudi Arabia pursued a deadly intervention in Yemen’s civil war.

Trump granted an enormous concession to Prime Minister Benjamin Netanyahu when he announced earlier this month that the United States would formally recognize Jerusalem as the capital of Israel. But he did not ask anything of Netanyahu in return.

That showed another hallmark of Trump’s foreign policy: how much it is driven by domestic politics. In this case, he was fulfilling a campaign promise to move the U.S. Embassy to Jerusalem from Tel Aviv. While evangelicals and some hard-line, pro-Israel American Jews exulted, the Palestinians seethed — leaving Trump’s dreams of brokering a peace accord between them and the Israelis in tatters.

Globalists vs. Nationalists

For some of Trump’s advisers, the key to understanding his statecraft is not how he deals with Xi Jinping or Angela Merkel, but the ideological contest over America’s role that plays out daily between the West Wing and agencies like the State Department and the Pentagon.

“There’s a chasm that can’t be bridged between the globalists and the nationalists,” said Stephen K. Bannon, the president’s former chief strategist and leader of the nationalist wing, who has kept his boss’ ear since leaving the White House last summer.

On the globalist side of the debate stand McMaster, Secretary of Defense Jim Mattis, Secretary of State Rex W. Tillerson, and Trump’s chief economic adviser, Gary D. Cohn. On the nationalist side, in addition to Bannon, stand Stephen Miller, the president’s top domestic adviser, and Robert Lighthizer, the chief trade negotiator. On many days, the nationalist group includes the commander in chief himself.

The globalists have curbed some of Trump’s most radical impulses. He has yet to rip up the Iran nuclear deal, though he has refused to recertify it. He has reaffirmed U.S. support for NATO, despite his objections about those members he believes are freeloading. And he has ordered thousands of additional U.S. troops into Afghanistan, even after promising during the campaign to stay away from nation building.

This has prompted a few Europeans to hope “his bark is worse than his bite,” in the words of Westmacott.

Trump acknowledges that being in office has changed him. “My original instinct was to pull out,” he said of Afghanistan, “and, historically, I like following my instincts. But all my life I’ve heard that decisions are much different when you sit behind the desk in the Oval Office.”

Yet some things have not changed. Trump’s advisers have utterly failed to curb his Twitter posts, for example. Some gamely suggest that they create diplomatic openings. Others say they roll with the punches when he labels Kim Jong Un “Little Rocket Man.” For Tillerson, however, the tweets have severely tarnished his credibility in foreign capitals.

“All of them know they still can’t control the thunderbolt from on high,” said John D. Negroponte, who served as director of national intelligence for President George W. Bush.

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