The Trump vs. Pelosi feud just got worse. Can Washington get anything else done?

WASHINGTON —  Any semblance of a working relationship between Washington’s two most powerful political leaders disintegrated perhaps beyond repair this week, leaving doubts about whether Congress and the White House will be able to accomplish anything in this election year.

In back-to-back public appearances the day after the Senate acquitted him in the impeachment trial, President Trump and House Speaker Nancy Pelosi (D-San Francisco) traded barbs that got remarkably personal, including Trump questioning the Catholic faith of Pelosi and the speaker saying the president is “talking about things he knows little about: faith and prayer.”

That followed an unusually public display of their mutual disrespect when Trump refused to shake Pelosi’s hand before the State of the Union address Tuesday and the speaker dramatically tore up of a copy of his speech from the dais after he had finished.

Trump and Pelosi paired their insults Thursday with offers to work together on their shared priorities, such as an infrastructure package and a bill to lower prescription drug costs.

But their caustic comments raised significant questions about where Congress and the White House are heading in the bitter partisan aftermath of impeachment, particularly during an election year when substantive legislative change is already hard to achieve.

“The coin of the realm in politics has always been a level of trust,” Rep. Mark DeSaulnier (D-Concord) said. “If you couldn’t trust someone, whether they’re Republicans or Democrats, or conservatives or liberals, these institutions don’t function.”

The House speaker suggested Thursday that her public displays of disapproval of Trump marked the turning of a page in her relationship with the president.

“I feel very liberated,” she said. “I feel that I’ve extended every possible courtesy. I’ve shown every level of respect.”

Even amid their war of words, however, Pelosi and Trump on Thursday left open the possibility of compromise or negotiation in the future.

And there could be strong incentives to do so.

“I think both sides have things that they need to get done that they can take to the voters to present as a record of accomplishment,” said Sen. John Thune (R-S.D.), predicting that time would probably soften the rancor.

“My impression around here is that everything has a pretty short shelf life,” he said. “What’s a crisis today tomorrow will be something else.”

Such bipartisan cooperation was evident last month when Congress passed a revised North American Free Trade Agreement. It had been a top campaign promise of Trump. But Democrats demanded significant changes at the end regarding labor and the environment. Now both sides claim the new U.S.-Mexico-Canada Agreement as a victory.

“There’s no such thing as an eternal animosity,” Pelosi said. “You never know of what cause you may come together with somebody you may perceive as your foe right now. Everybody is a possible ally in what may come next.”

Perhaps the most likely place for common ground would be an infrastructure package — funding to rebuild bridges and roads, expand broadband to rural areas and build water systems, among other initiatives.

“I hardly ever had a conversation with [Trump] when he wasn’t talking about infrastructure, infrastructure, infrastructure,” Pelosi said. “I think he really wants that.”

Toward the end of a speech Thursday, Trump also referenced what could be accomplished together.

“What we can do working with both parties in Congress is — would be unbelievable,” he said. “Think what we could have done if the same energy was put into infrastructure, prescription drug prices. Think of what we could have done.”

But there are other reasons to be skeptical that Washington can even accomplish an infrastructure package. For years, it has been the first item politicians name when asked about bipartisanship, but one that never seems to move forward.

Congressional Democrats and Trump had begun negotiations on an infrastructure package in May. But talks were scuttled when Trump stormed away from a White House meeting, arguing that compromise was impossible in the face of House oversight investigations.

Rep. Peter A. DeFazio (D-Ore.), who as chairman of the House Transportation Committee is writing an infrastructure bill that he hopes to release in March, appeared unfazed by the idea that Trump — or Pelosi — is interested in reengaging on the issue.

He said he hasn’t heard from the White House since before a senior legislative aide left in May last year. “I don’t even know who the liaison at the White House would be,” he said.

And then there’s the open question of whether Trump and Pelosi could actually work together post-impeachment.

Pelosi on Thursday called his State of the Union address “a reality show,” “quite appalling,” “beneath the dignity of the White House” and “a manifesto of mistruths, of falsehoods, blatantly, [and] really dangerous to the well-being of the American people.” She ended her Capitol Hill news conference suggesting that he might have been “sedated” as he delivered the speech.

In the White House speech to tout his Senate acquittal, Trump said “Nancy Pelosi is a horrible person” and questioned whether “she prays at all,” let alone for him, as she has said she does. He called Democrats “vicious people” who would have impeached George Washington.

Pelosi took particular umbrage at Trump’s criticism of her religious faith and that of GOP Sen. Mitt Romney of Utah, who was the only Senate Republican to vote to remove Trump from office Wednesday. Romney said his personal faith compelled his vote.

At the National Prayer Breakfast on Thursday, Trump said: “I don’t like people who use their faith as justification for doing what they know is wrong, nor do I like people who say, ‘I pray for you,’ when they know that’s not so.”

The tension between Pelosi and Trump has gotten particularly acute since Democrats took control of the House in early 2019, elevating Pelosi to one of the most powerful positions in Washington.

Earlier in Trump’s term, he seemed to have a measure of affection and respect for the House Democratic leader. Though she seldom publicly conveyed any sense of warmth toward him, he spared her — for a while — from his nearly constant public lashing on Twitter. He didn’t attack her even when she compared his quest for a border wall to a “manhood thing.”

But the relationship — as tenuous as it was when Democrats were in the minority — would not survive 2019. The back-and-forth was minor at first. He dubbed her “Nervous Nancy.” She postponed his 2019 State of the Union address so it would not take place during the government shutdown. In response, he canceled her planned trip with a congressional delegation to Brussels and Afghanistan.

The two have not spoken since an October White House meeting that ended with Pelosi walking out after he called her a “third-grade” politician and she told him “all roads with you lead to Putin.”

Times staff writer Eli Stokols contributed to this report.