Mr. Trump said he still intended to push for the wall, and Republicans sought to minimize the depth of his agreement with the opposition. But there was no escaping the fact that twice in a week’s time Mr. Trump embraced the top Democrats while keeping his supposed Republican allies well at arm’s length.

Given turnover on Capitol Hill, many lawmakers simply haven’t experienced this dynamic, in which the president of one party tries to work amicably with leaders of the other. And the chaos has been intensified by the fact that the two sides have given slightly different versions of events — a development usually avoided when one party or the other strictly controls the message.

The talks between Mr. Trump and the Democrats haven’t been the only tentative stabs at bipartisanship. Republicans and Democrats on the Senate health committee have been meeting to try to work out a consensus way to shore up the struggling health insurance exchanges. Turnout has been strong and the talks productive, participants said.

“We had more than 60 senators who came to either the meeting with the witnesses before the hearing or the hearing itself and engage in a serious discussion,” said Senator Lamar Alexander, the Tennessee Republican who leads the committee. “I haven’t seen anything like that since I came to the Senate 15 years ago.”

Knocked off balance, Republicans say they really have no choice but to find a way to navigate within the president’s new architecture. They say it would have been more truly bipartisan had Mr. Trump invited Republican leaders to join him at the dinner table with the Democrats. And they don’t expect all that many occasions when Mr. Trump and Democrats are able to come together.

Senator Mitch McConnell, the Kentucky Republican and majority leader, said that on most topics, Mr. Trump was still more naturally aligned with Republicans.