WASHINGTON —  They tend to gush over each other in public, but their private dealings are often opaque. Both are brazenly transactional, and cling tenaciously to grudges. Each likes to keep everyone around him guessing.

Of all President-elect Donald Trump’s relationships with world leaders — which are coming into sharper focus as he prepares to take office again in less than two months — that with Russian President Vladimir Putin may be the most consequential, and the most fraught.

Trump says his foreign policy motto will be “America first.” Critics fear that Trump will be steamrolled by the former Soviet intelligence officer on Ukraine, on sanctions aimed at curbing Russian aggression, and on the future of the North Atlantic Treaty Organization.

Whatever his course, he now has a better grasp of the levers of power in the administration he will soon lead than he did in his first term.

“I think maybe Trump has a better idea now of how to be president,” said Kadri Liik, a senior policy fellow at the European Council on Foreign Relations, discussing Trump’s past dealings with Putin. “So let’s see.”

Throughout Trump’s first term in office, his ties to Putin offered an odd leitmotif: the episodes of striking public deference to the Russian leader, Trump’s often-stated assertion that a personal bond with Putin benefited the United States rather than undermining it, his unabated fury over the lengthy federal investigation of Moscow’s interference in the 2016 presidential election on Trump’s behalf.

Back in 2018, at a joint news conference by the two leaders in the Finnish capital of Helsinki, Trump’s declaration that he believed Putin over his own intelligence agencies so alarmed a then-advisor, the Russia scholar Fiona Hill, that she later recounted being tempted to feign a health emergency or pull a fire alarm to stop him.

In Trump’s between-terms interregnum, he and Putin seemingly stayed in touch, with at least seven phone conversations that took place outside the purview of U.S. diplomacy, according to journalist and author Bob Woodward.

This time around, Trump inherits the war in Ukraine, a conflict to which the United States is not a party, but which Putin paints as a potentially direct confrontation with any Western military partner of the Kyiv government. He will also face a loose axis of adversarial powers in which Russia is a junior player to China, but bolsters Moscow’s avowal that a U.S.-led world order has ended.

The Trump-Putin relationship over the next four years could help determine how that axis tilts.

After Trump won November’s election, he and Putin enacted what some analysts described as an elaborate set piece that encompassed elements of both conciliation and jockeying for dominance.

The Russian leader offered up his first public congratulations on Trump’s election win somewhat offhandedly, in a question-and-answer session that followed a lengthy speech. But at the same appearance, he volunteered a compliment guaranteed to endear him to the president-elect, praising Trump’s “manly” reaction to a July assassination attempt that left the then-candidate with a minor ear injury.

Then came an odd kerfuffle over who might have called whom: The Trump team let it be known there had been a congratulatory phone call, but the Kremlin then denied news reports of it. Around the same time, a widely watched program on Russian state TV aired decades-old explicit photos of former and soon-to-be First Lady Melania Trump, a onetime model, while its hosts openly smirked.

Almost immediately after the election, there was a much-parsed comment from Nikolai Patrushev, a Putin aide, who gave an enigmatic reply when asked what Trump’s win portended for Russia.

“To achieve success in the election, Donald Trump relied on certain forces to which he has corresponding obligations,” Patrushev told the business publication Kommersant, in remarks that were amplified by the official Tass news agency. “As a responsible person, he will be obliged to fulfill them.”

While ambiguous, the comment was read by some observers as positing that Trump was somehow beholden to Moscow — but was also typical of a sly, suggestive style often employed in Kremlin propaganda.

Trump, for his part, spent the last few weeks unveiling a series of Cabinet picks that included some notable Russia skeptics, at least in their previous incarnations.

But for one particularly crucial post — the director of national intelligence, who oversees 18 U.S. intelligence agencies that gather and safeguard the nation’s most closely held secrets — he picked Tulsi Gabbard, a former congresswoman whose stated pro-Kremlin views have raised concerns even among Trump’s fellow Republicans.

“The appointment to such a sensitive role of someone with so many questions around them, whose nomination has been welcomed on Russian TV, escalates the concern that many observers have,” Ruth Deyermond, a senior lecturer in post-Soviet security at King’s College London, wrote in an email.

The choice of Gabbard, she said, “confirms existing signals that point to a very pro-Russian White House.”

An early test is likely to be Ukraine. There is a broad expectation that Trump will seek to leverage a threatened weapons cutoff into a deal that might force the government in Kyiv to give up Russian-captured territory and renounce aspirations to join NATO.

On Wednesday, Trump unveiled his choice of a special envoy for Russia and Ukraine: Keith Kellogg, a retired three-star general. A staunch conservative and an advisor in the first Trump administration, he has pushed for a plan under which Ukraine would have to cede some territory to end the war.

But Putin might not be positioned to get his way entirely. Liik, of the European Council on Foreign Relations, said the Russian leader, in seeking to make Ukraine a “vassal state,” could overreach.

“Putin wants more than Trump is ready to offer,” she said. “I’m not sure Trump is ready to go to those lengths, if it makes him look like a loser.”

Still, the president-elect can effect profound changes in the security order even if he does not follow through on every implied threat.

During his first term, Trump routinely denigrated the North Atlantic Treaty Organization, and suggested as a candidate that he would let Russia do “whatever the hell they want” to European allies he thought were ducking defense-spending obligations.

“I don’t expect Trump to formally withdraw the U.S. from NATO, but his team’s words and actions to date have already weakened it,” said Deyermond.

Some observers, though, say that with regard to actual policymaking, overall Russian expectations for the coming Trump presidency may be relatively low. After Trump’s first win, in 2016, “pro-Kremlin propaganda mouthpieces openly crowed that the victory was really Moscow’s,” said Alexander Baunov, a senior Eurasia fellow at the Carnegie Endowment for International Peace.

“This time around, things are very different,” he wrote in the digital publication Carnegie Politika. “The jubilation in Moscow is far more muted.”

This week brought an acerbic assessment of both Trump’s and Putin’s personality traits from none other than former German Chancellor Angela Merkel, who wrote in a newly published memoir about her dealings with both of them.

Merkel, who stepped down in 2021, described Trump as “clearly fascinated” by the Russian president, adding that he seemed “captivated by politicians with autocratic and dictatorial traits.”

Putin, on the other hand, was “someone who was always on guard not to be treated badly, and always ready to dish out punishments” — including once bringing a large black Labrador to a meeting with Merkel, knowing she was afraid of dogs.

Describing one of her encounters with Trump, Merkel called him “emotional.” But she counseled that a calm, dispassionate approach worked better with someone like Putin.

“You could find all this childish, reprehensible; you could shake your head at it,” she wrote of Putin’s manipulative style. “But that didn’t make Russia disappear from the map.”