Donald Trump, right, welcomes pro-Brexit British politician Nigel Farage at a campaign rally in Jackson, Miss., in August. Farage says he will attend Trump's inauguration. (Gerald Herbert/AP)

With the transatlantic relationship on the line, European leaders are trying to put a brave face on Donald Trump’s new world order.

They may hold the American president-elect in profound disdain, a feeling many haven’t bothered to conceal. But through gritted teeth, they insist that the ties anchoring the globe since World War II will endure — if not with much warmth, then at least through the sort of transactional relations Trump can understand.

That theory, however, will be put to the test as tweets and interviews turn into policy and action.

After decades of transatlantic relations based on a shared set of interests and values, Europe is reckoning with what could happen if only one of those remains.

“It’s where interests and values intersect where we’re going to find problems,” said Robin Niblett, director of the London-based think tank Chatham House. “If our values stand for anything, it’s open, democratic societies and open markets. If America moves away from those, that’s pretty fundamental.”

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Trump supporters would undoubtedly say he is fully committed to democracy and capitalism. But many Europeans view in his tweets and raucous rallies a fundamental shake-up of the values underpinning the liberal international order.

Analysts and former European officials say the list of potential flash points includes Russia, Iran, Israel and Palestine, climate change, democracy promotion and global trade.

To European leaders, the gap goes beyond mere policy differences.

The president-elect’s determination to embrace adversarial autocrats such as Russian President Vladimir Putin even as he pointedly criticizes allied democrats such as German Chancellor Angela Merkel; his threats to discard internationally agreed deals such as the Paris climate accord or the Iranian nuclear agreement; his indifference to the fate of the European Union and dismissive approach toward NATO; his apparent lack of concern about evidence of Russian interference in the U.S. election; and his oft-repeated jabs at free trade and the media all point to an Atlantic-size gulf between the new U.S. commander in chief and the European establishment.

Serious tensions have erupted between Europe and Washington before — think the “freedom fries” era, when France and Germany rejected the Bush administration’s march to war in Iraq. But rarely have Europeans felt that fundamental values may be so deeply in opposition.

“We’re getting into uncharted territory here, where Europeans will have to strike a balance between a transatlantic relationship that remains intense and a deep fundamental disagreement on values and an interpretation of what democracy is about,” said Stefano Stefanini, a former senior Italian diplomat who was his nation’s ambassador to NATO from 2007 to 2010.

It is difficult to know to what extent Trump will carry through on his foreign policy promises. In confirmation hearings, his cabinet nominees have at times sharply disagreed with the man who selected them. Analysts also note that Trump’s promises are often contradictory.

But European nerves were set on edge anew this week when Trump told interviewers from a British and a German newspaper that he thought more countries would follow Britain out of the E.U., that NATO was “obsolete” and that Merkel had made “a catastrophic mistake” by welcoming hundreds of thousands of refugees.

The sudden unpredictability of Europe’s most important global partner comes as the continent is deep in the throes of its own identity crisis, reeling from the same populist shocks that brought Trump to power.

Brexit, the fall of Italy’s center-left prime minister and the emergence of strongly nationalistic governments in Eastern Europe are all considered symptoms of the anti-establishment tide washing over the Western world.

Yet unlike in Washington as of Friday, the establishment remains in charge in most European capitals.

But no one knows for how long.

With elections looming this year in the Netherlands, France and Germany, far-right parties are making a concerted push to end the centrist consensus that has prevailed in Western Europe for generations. That dynamic, coupled with Trump's unpredictability, makes it almost impossible to say what contours the U.S.-European relationship will take.

“There’s absolutely huge uncertainty over what kind of U.S. administration Europe will find itself dealing with,” said Adam Thomson, director of the European Leadership Network and a former British ambassador to NATO. “There’s also uncertainty for the Trump administration over what kind of Europe America will be dealing with.”

For the most part, European leaders have tried to project confidence that the Trump era will be business as usual, with shared interests in combating terrorism, tamping crises and promoting economic growth overriding any differences.

Britain, with one foot out the door of the E.U., has been particularly keen to cultivate close ties and ensure its “special relationship” with Washington remains intact.

Europe’s accommodating response reflects how internally divided it has become, and how little power it has relative to Washington. The United States spends vastly more on defense than Europe — nearly three times as much as all European members of NATO combined.

Trump has demanded that NATO allies pay their own way, while raising questions about whether he would come to members’ defense in the event of an attack. His closest European ally is neither British prime minister Theresa May nor Merkel, but Nigel Farage — the bomb-throwing Brexit champion who wants Britain’s departure to be the trigger event in the E.U.’s ultimate collapse.

“There has never been an American president who did not support European integration. It’s a first. A tragedy for Europe,” Stefanini said.

Despite European division and weakness, leaders could be left with little choice but to distance themselves from Trump if he follows through on pledges considered antithetical to European values.

A ban on Muslim immigration, a resumption of the use of torture or an end to American participation in the Paris climate accord — all Trump campaign promises — would undoubtedly elicit strongly negative reactions in Europe.

But perhaps most critical will be his handling of two of the West’s adversaries: Russia and Iran.

Trump appears determined to improve ties with Putin, just three years after Russian military intervention in Ukraine prompted the United States and Europe to impose sanctions.

In theory, much of Europe would welcome a lowering of tensions between Russia and the West, said Thomson, the former British ambassador. But the details will be critical.

“It’s important for Europeans that this is not seen as some kind of sellout to Russia, and that there’s not a U.S.-Russia deal done over European heads,” Thomson said.

On Iran, too, the maneuvering will be extremely delicate. Trump has repeatedly attacked as “a really, really bad deal” the nuclear agreement negotiated in 2015 between Iran and six of the world’s leading powers — the United States, the United Kingdom, France, Germany, Russia and China.

Any Trump move to unilaterally pull the United States out risks antagonizing every other party to the agreement. Critics say it could also undermine global faith in Washington’s commitment to live up to its promises.

If Europe is going to effectively counter Trump, however, it will have to stay united. And it is far from clear that it can.

The most powerful voice in Europe belongs to Merkel, who delivered a tough message to Trump after his election win. Her barbed congratulatory note said she looked forward to working together with him based on “common values — democracy, freedom, as well as respect for the rule of law and the dignity of each and every person.”

The declaration was a signal not only to Trump but to her own voters ahead of an election in which she faces the likelihood of a far-right anti-immigrant party capturing seats in the Parliament for the first time.

“A lot of thinking in Berlin is already going into how we will have to reckon with Trump in actions he takes strategically or actions he takes that impact the Western liberal order,” said Daniela Schwarzer, head of the German Council on Foreign Relations.

But there are limits to how far Germany can go.

“Even if Germany is willing to take the lead, it needs to work with others. And this is a messy picture,” Schwarzer said.

With Britain focused on its E.U. exit, London is unlikely to take a stand against Trump unless it is forced to do so. Europe’s other major power, France, has its own struggles as it prepares for spring presidential elections in which a far-right party stands a chance of victory.

Whoever wins, French policy analysts said they feared their country was entering a new era of ties with Washington.

“In many ways, it’s not so much what America will do but, in a way, what America has become,” said Dominique Moisi, a co-founder of the French Institute for International Relations. “It’s essence as much as it’s performance. We don’t know what the performance is going to be, but we have an inkling that the essence of America has changed.”

Birnbaum reported from Brussels and McAuley from Paris.