British Prime Minister Theresa May will visit President Trump on Jan. 27, 2017. (European Pressphoto Agency; Matt McClain/The Washington Post)

When British Prime Minister Theresa May on Friday becomes the first world leader to meet with President Trump in the Oval Office, the two will have much in common to discuss.

Both were catapulted to power on the back of populist shocks in 2016. Both have promised to deliver radical change to their countries. And both now lead nations at the heart of a Western alliance facing its most serious identity crisis in decades. 

Yet beneath the similarities lie profound differences in style and substance that make the two leaders less the second coming of the Thatcher-Reagan transatlantic lovefest and more a geopolitical odd couple. 

May is everything that Trump is not: a careful, low-key and pragmatic member of the political establishment with a decades-long career in elective office. She holds mainstream positions on critical issues such as trade and security that put her sharply at odds with Trump’s protectionism and isolationism. She tweets about Christmas and World AIDS Day, not alleged voter fraud or feuds with the news media, celebrities or political opponents.

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She prizes the NATO military alliance and holds skeptical views of Russia — uncertain ground with Trump.

Whether those differences dominate their meeting or they manage to bond over their shared circumstances, this could be a critical moment for both leaders.

May probably has more to gain or lose from the visit, which includes an unusual address to Republican members of Congress the day before she visits the White House.

May will say Brexit offers a chance for a new partnership with the United States, according to excerpts of her remarks to Republicans released Wednesday.

“As we rediscover our confidence together — as you renew your nation just as we renew ours — we have the opportunity — indeed the responsibility — to renew the special relationship for this new age,” May will say. “We have the opportunity to lead, together, again.”

With her country preparing to leave the European Union, she is gambling her premiership on her ability to forge new ­relationships beyond the continent — with a strengthened ­Anglo-American bond at the top of her wish list. She also needs to convince Trump that NATO fits into his “America first” vision of defense and overseas engagement.

The invitation to be Trump’s first foreign visitor is a diplomatic nicety that was in doubt as recently as a week before the visit. And it follows an awkward series of actions by Trump that could easily be read as snubs. Just days after Trump’s win, he invited anti-E.U. firebrand Nigel Farage to meet him at Trump Tower — then tweeted that Farage would make a fine British ambassador to the United States. According to a leaked transcript, Trump suggested during their first telephone call that if May were passing through Washington, she should let Trump know.

The breach of diplomatic protocol alarmed May’s inner circle, which sees Farage as a meddlesome adversary. Rather than jabbing back, however, Downing Street began trying to curry favor. 

Foreign Secretary Boris Johnson boycotted an emergency E.U. meeting called to discuss Trump’s win, dismissing it as a “whinge-o-rama.” May also made a point of rejecting Obama administration criticism of Israel, echoing Trump talking points.

British officials have even gone so far as to float the idea that May could play Thatcher to Trump’s Reagan in a revival of 1980s-style relations between Washington and London. 

British political observers say, however, that the sort of transatlantic warmth exhibited at the height of the Cold War is unlikely with this duo because of a fundamental mismatch in worldviews. 

Their approach toward Vladimir Putin’s Russia — May is a hawk while Trump says he wants closer ties — is just one glaring example.

“The problem is that Ronnie and Maggie had a common enemy in the Soviet Union and world communism,” said Tim Bale, a politics professor at Queen Mary University of London.

On a range of other issues — global trade, NATO, climate change and the nuclear accord with Iran — there’s a similarly wide gap between the two leaders. 

For Trump, May’s arrival within a week of his inauguration is a chance for him to project legitimacy and normality to U.S. allies still struggling to come to grips with his unexpected electoral victory and who still view him with suspicion — if not outright contempt. The meeting suggests a continuity with familiar American foreign policy priorities that also reassured some Trump skeptics at home.

The “special relationship,” as the modern U.S. alliance with Britain is called, has always been an unequal one, with the United States the richer and more powerful partner.

But the partnership has huge benefits for the United States, especially in the intelligence, diplomatic and military realms. The two countries share intelligence closely and usually move in tandem in international negotiations such as the Iran nuclear pact. The United States has relied on Britain’s highly trained armed forces in Iraq, Afghanistan and Ukraine, sometimes to the domestic political detriment of British leaders.

Trump, a self-proclaimed Anglophile, has signaled a willingness to negotiate a free-trade deal with Britain as soon as it is out of the E.U. British officials have responded with enthusiasm, and May has said the issue will be at the top of her priority list for her meeting with Trump. 

The meeting, she told Parliament on Wednesday, is “a sign of the strength of the special relationship between the United Kingdom and the United States of America. A special relationship on which he and I intend to build.”

Yet British analysts say May is deluding herself if she thinks Trump is the partner Britain needs to ensure its safe landing outside the E.U. 

“If you look at the way Donald Trump thinks about deals, rushing over there as quickly as possible and looking like you’re desperate doesn’t exactly work to your advantage,” said Mark Leonard, director of the European Council on Foreign Relations. “So I don’t think this is very clever of her from a tactical standpoint.”  

Unlike other European leaders, such as German Chancellor Angela Merkel and French President François Hollande, who have held Trump at arm’s length and emphasized that their friendship is contingent on Trump’s not trampling core Western values, May has rushed to embrace the new U.S. president. 

Bale said that one reason May probably wanted to meet Trump early in his presidency is that she wanted to find out for herself where he stands when he’s not whipping up a crowd or provoking opponents on Twitter. She may even think that on some issues she can bring him around to her views. 

But the diplomacy will be exceptionally tricky.

She cannot afford to antagonize the famously thin-skinned Trump, because she needs his support for a trade deal. But if she does not challenge him, Bale said, then “she’ll be seen to be sucking up to someone who shouldn’t be sucked up to and who can’t be relied upon. That could backfire at home, and it could do damage to her relations with other European leaders.”

Gearan reported from Washington. Karla Adam in London and Carol Morello in Washington contributed to this report.