WASHINGTON — For all the talk of a Tea Party of the left, the true power in the House revealed its face last week — the Mighty Moderates.

The failure of House liberals to attach strict conditions to billions of dollars in emergency border aid requested by President Trump highlighted the outsize power of about two dozen centrist Democrats, mainly from Republican-leaning districts, who are asserting themselves to pull the chamber to the right.

Their views diverge sharply from the mostly liberal cast of Democrats seeking their party’s presidential nomination and from the diverse new crop of outspoken liberals in the House who have captured the public’s imagination and infused new energy into the progressive majority of their caucus. But their victories in districts that Mr. Trump won in 2016 are the reason Speaker Nancy Pelosi of California, a staunch liberal herself, holds her gavel. For now, they are proving far more effective at wielding their influence than those in the party’s vocal and headline-grabbing left.

“Where they come from is crucial to holding the House, and that makes them very influential,” said Laura Hall of the Bipartisan Policy Center in Washington. “If you’re Speaker Pelosi or another member of the Democratic leadership, you have to always be thinking about those members whose seats went from red to blue and helped to flip the House.”