Taking stock of the feud, each side insisted they got the upper hand in a fight that shows no sign of waning 18 months before the 2020 elections, with implications for the economy as the budget and federal borrowing limit remain unresolved while the dispute regarding oversight between the White House and Congress rages.

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Pelosi’s allies said she showed up the president and reinforced an image of a chief executive behaving so badly and childishly that he is unfit for office — a clear message to voters next year. But to Trump’s backers, the president succeeded in highlighting that an already unpopular politician is struggling not only with the far-left liberals in the Democratic ranks, but even some on her leadership team.

“She has very challenging dynamics in her conference, and she’s trying to appease her conference,” said Marc Short, chief of staff to Vice President Pence. “She has a very difficult job.”

Some White House aides also said it was better for Trump to be fighting with Pelosi than former vice president Joe Biden or other 2020 candidates, as the president has done recently, elevating their status.

For Trump and Pelosi, the series of salvos was a break from past practices. Trump has derided other politicians with nicknames, but refrained from mocking Pelosi, the most powerful woman in Democratic politics. During the 2018 midterm elections, the speaker had instructed Democratic candidates to focus on health care, education and other issues rather than on Trump, who was not on the ballot. The strategy, which she also adopted, paid off as the party reclaimed the House majority.

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Pelosi’s allies said her taunting of Trump now is intentional, designed to get under his skin and elicit an angry reaction, according to officials close to her who spoke on the condition of anonymity to discuss private deliberations. She contended that his resistance to investigations was to goad her members to back impeachment, which would undermine her party.

Emerging from a special closed-door caucus meeting on Capitol Hill on Wednesday, Pelosi took the unusual step of speaking to reporters and in front of television cameras, accusing Trump of “engaging in a coverup” in response to congressional subpoenas.

At the White House a short time later, Trump angrily walked out of a meeting with Pelosi and other Democrats on infrastructure after three minutes. The president told reporters in remarks in the Rose Garden that he could not work with Democrats until they “get these phony investigations over with” and argued that special counsel Robert S. Mueller III’s nearly two-year investigation had cleared him of any wrongdoing.

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Back on Capitol Hill, Pelosi and the Democrats kept up their criticism, with the speaker suggesting Trump’s possible “lack of confidence” prevented him from reaching a deal on infrastructure and saying she prayed for the president.

Pelosi wrote of Trump’s “temper tantrum” in a letter to colleagues Wednesday afternoon, and the next day she was relentless in her attacks, suggesting his White House aides and family “should stage an intervention for the good of the country.”

Hours later, Trump called her “crazy Nancy” at a White House event on aid to farmers, impugned her mental clarity and intelligence and pressed aides to attest to his calmness during the meeting the previous day. He later tweeted a spliced video that made her appear confused.

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White House aides say Trump was more frustrated by the “coverup” comment than her Thursday commentary likening him to a toddler. He flew into a rage Wednesday morning after she made those remarks — and then stewed as she continued to taunt him from Capitol Hill. The aides spoke on the condition of anonymity to discuss private talks.

Pelosi’s allies insist the events left Trump as the one to blame for the failure to reach a deal on infrastructure, as he had taken responsibility in advance for the government shutdown in an Oval Office meeting with Pelosi and Senate Minority Leader Charles E. Schumer (D-N.Y.) in December.

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When the shutdown — the longest in history — ended earlier this year after 35 days, Pelosi was seen as the winner in the standoff.

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“The speaker knows how to use power,” said longtime Pelosi ally Rep. Anna G. Eshoo (D-Calif.). “She knows who. She knows how. She’s a master negotiator. . . . I think the president is really befuddled by her.”

Her allies argued that his angry reaction feeds the narrative of an erratic president.

Trump’s proponents pushed back.

“She talks about him like he’s incompetent. It’s totally ridiculous,” said Rudolph W. Giuliani, the president’s lawyer. “You may not like him, you may despise him, but there’s no question he is mentally and physically capable to do the job.”

As for Pelosi, Giuliani said she is “not exactly the most articulate person in the world. The last couple of weeks, she’s been talking funny. I’ve noticed it, and a lot of other people have noticed it.”

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Giuliani shared a doctored video this week showing Pelosi slurring her words but deleted it after significant blowback.

Until recently , Trump was telling advisers that he wanted to reach a deal on infrastructure and had even talked to the trucking industry about a gas tax to help finance upgrades to the nation’s roads, bridges and tunnels. He also telephoned Pelosi to tell her how good her television coverage was after the two of them huddled to discuss infrastructure three weeks ago.

Trump, who has long admired Pelosi and showered her with compliments for her grip on her caucus, also recently told West Wing aides how tough she is — and how she keeps her party in line with an iron fist. “She has some real crazies,” he said recently to an adviser, who spoke on the condition of anonymity to discuss private conversations.

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While Pelosi has employed a strategy of trying to make Trump appear childish, she actually has told colleagues she thinks he’s unworthy of such a comparison. When a Democrat compared Trump to a fifth-grader, Pelosi responded that such a remark was an insult to fifth-graders, according to an individual who spoke on the condition of anonymity to describe the conversation.

“Don’t say that. Children are wonderful!” said the mother of five and grandmother of nine.

Privately, Pelosi repeatedly has said Trump is not worth impeaching. She has managed to tamp down the clamor for impeachment from an increasing number of Democrats, arguing that Trump would welcome the move and an acquittal vote in the ­Republican-led Senate.

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For now, the talk of impeachment has quieted, even as Pelosi accused Trump of a “coverup.”

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Trump allies predicted she was trying to appease her frustrated caucus and base.

“She was trying to throw a little red meat to her caucus right before they headed home to the long recess,” said David Urban, a Trump ally who worked on the 2016 campaign and the GOP convention. “And Pelosi, who is usually a masterful politician, misjudged the president’s reaction. She made a statement that overplayed her hand. Now we’re going to be at some place of an impasse we haven’t seen before.”