It has long been an axiom of the House majority: Vote against a piece of legislation put forth by your party if you absolutely must, but never, ever vote against the “rule” to bring that legislation to the floor.

Until the last few weeks, that standard had held for more than two decades. But now, about a dozen rebellious House Republicans have decided to leverage their badly needed votes on the routine procedural measures to win policy concessions, breaking the longstanding code of party discipline and threatening the traditional operation of the House.

“Who cares?” asked Representative Eli Crane of Arizona, one of the members of the far-right House Freedom Caucus who broke with their party on the rule last week, resulting in a very rare defeat that ground the chamber to a halt for a week.

“I could care less, and neither could my voters,” he said about violating a House norm. “Tradition be damned; we have to change the way this town works.”

But making such changes would fundamentally alter the nature of the House. Unlike the Senate, where consensus is required to move forward on virtually anything, the House operates according to strict majority rule, making party unity critical to the smooth functioning of business.

It could also greatly complicate life for Speaker Kevin McCarthy, with potentially disastrous consequences for the country. With a razor-thin margin of control, if all Democrats are present and voting “no,” he can afford to lose no more than four Republicans and still win approval of the rules that are required to bring most major legislation to the floor.

Should the group of Republicans continue their procedural resistance, it would sow chaos for Mr. McCarthy’s efforts to legislate, potentially upending the consideration of government spending bills needed to avert a shutdown this fall. Their new strategy gives the mutineers, who lack the votes to oust Mr. McCarthy, significant power over the speaker and his agenda.

They were outraged over Mr. McCarthy’s deal with President Biden to suspend the debt limit and avert a federal default, which included spending caps far higher than what House Republicans had endorsed.

Mr. McCarthy met with the rebels on Monday and appeared to quell the procedural mutiny at least temporarily by offering new assurances, and the House got back to business on Tuesday. But the insurgents said they had explicitly refused to promise to vote for rules in the future, and would have no problem defeating them in the weeks ahead should Mr. McCarthy fail to share power with them in the ways they demanded.

“Each day there’s another challenge,” Mr. McCarthy said. “I just wake up every day, pray for the patience of Job and find a solution. We work forward.”

While little noticed by the public, House rules are fundamental to the functioning of the chamber. They put the agenda firmly in the hands of the majority, allowing the party in power to control what legislation is considered and what changes may be proposed. Past speakers of both parties have made supporting the rules a strict test of party loyalty, and lawmakers would defect on them at risk of punishment such as losing committee posts or having their legislation bottled up.

But hard-right conservatives suspicious of Mr. McCarthy’s conservative bona fides have had the usually obscure rules process in their sights since he had to fight for his speakership in January. Among the concessions he made was an agreement to put three of the most conservative members of the House on the Rules Committee, usually an organ of the speaker.

The committee narrowly sent the debt limit legislation to the floor late last month over the objections of two Freedom Caucus members on the panel. But more than two dozen Republicans opposed the rule to allow the compromise to be considered, forcing Mr. McCarthy to rely on Democrats to cross party lines and rescue the bill. It was an extraordinary turn that underscored the speaker’s weak grip on his conference, and angered conservatives even more.

Democrats were not about to help Republicans last week when 11 far-right conservatives opposed the rule on a political messaging bill about gas appliances that had no chance of becoming law. It was the first time since 2002 that a rule had been defeated, and it stunned the House, leaving Mr. McCarthy unable to put any bills on the floor. Lacking a governing majority as the hard right refused to relent, he sent lawmakers home for the week.

In a series of meetings after the floor revolt, Mr. McCarthy and the Freedom Caucus members sought to find some accommodation, with the far-right Republicans demanding more say in determining what bills reach the floor, particularly the spending bills that are beginning to take shape.

In trying to mollify the rebelling lawmakers, Representative Kay Granger, Republican of Texas and the chairwoman of the Appropriations Committee, said on Monday that the panel would keep spending bills below the amounts agreed to in the debt limit deal, a position certain to provoke a clash with Democrats.

The situation left Democrats declaring that the speaker had ceded control of the floor to what Representative James P. McGovern of Massachusetts, the senior Democrat on the Rules Committee, called the right wing’s extreme right wing.

“In what world should the 11 most extreme people in the Republican Party get to dictate the entire agenda of a legislative body that represents 332 million people?” Mr. McGovern asked.

The uproar left many other House Republicans frustrated, complaining that a handful of their colleagues were turning the chamber upside-down and worried about the electoral ramifications in 2024. They called for calm and said they hoped that opposing rules did not become as routine as supporting them once was.

“This can’t be the new normal,” said Representative Dusty Johnson, Republican of South Dakota. “We can’t afford to have this kind of disruption every week.”