“What we need is a speaker who follows the House rules,” Rep. Justin Amash (R-Mich.) said about the speaker race roiling the GOP. (Carlos Osorio/AP)

When the 40 or so Republican lawmakers responsible for the recent upheaval in the House talk about what it would take to quell their rebellion, they do not necessarily talk about the debt ceiling, the federal budget or any other demand of the party’s energized conservative base.

They speak instead about rule changes, committee assignments and the hallowed pursuit of “regular order” — a frequently invoked, civics-textbook ideal by which legislation bubbles up through subcommittees to committees to the floor to the president’s desk and into law.

“The false, lazy narrative is that we want a more conservative speaker,” Rep. Justin Amash (R-Mich.) told reporters at a forum of hard-line House members last week. “But the reality is: What we want is a process-focused speaker. . . . What we need is a speaker who follows the House rules.”

Those lawmakers, most of whom have organized into the House Freedom Caucus, are at the core of the leadership crisis afflicting the House GOP, and there are few signs they will retreat in the wake of Majority Leader Kevin McCarthy’s withdrawal from the speaker’s race.

Rep. Raúl R. Labrador (R-Idaho), said Friday, “The only way you’re going to bring people together here in the House is by changing the rules or at least following the current rules that we have.” (Bill Clark/CQ Roll Call via AP)

They want more bills and amendments from rank-and-file members and a generally more freewheeling approach on the House floor, as well as an overhaul to internal party-management rules. And, they say, they are willing to use the same leverage they used against McCarthy (R-Calif.) to get it.

The attention of the political world on Saturday remained trained on Rep. Paul Ryan (R-Wis.), who has been touted throughout the Republican Party as a consensus choice for speaker and who is reconsidering calls to run. But even Ryan wouldn’t be exempt from those demands, some hard-liners say.

“The only way you’re going to bring people together here in the House is by changing the rules or at least following the current rules that we have,” Rep. Raúl R. Labrador (R-Idaho) said Friday.

The problem with that, other Republicans say, is that those demands would make the House even more ungovernable than it already is.

“Everyone tries regular order, and nobody succeeds at it,” said John Feehery, who served as an aide to former speaker J. Dennis Hastert of Illinois. “What will end up happening is that conservatives will lose, because they don’t have the votes.”

The most frequently invoked breaches of “regular order” include the spending bills that have kept the government open for the past four years. Those bills, often written in crisis, have rarely been subject to formal committee votes and have passed on the floor with a coalition of roughly 50 centrist Republicans plus most Democrats.

But calls for regular order have also become a code for other, deeper frustrations, which have turned the debate about the future of the GOP leadership into one that is more about personalities than procedures.

“If you ask the 435 people in there what regular order is, you get 435 different answers,” said Rep. Bill Flores (R-Tex.), chairman of the Republican Study Committee, who agrees that some rule changes are needed.

The procedural demands, regardless of the semantics, played a significant role in McCarthy’s decision not to seek the speaker’s chair. In the days leading up to his announcement Thursday to withdraw from the race, it became clear that although McCarthy could cobble together enough Republicans to win a majority in a floor vote of all 435 members, doing so would involve making promises that would make the GOP caucus almost impossible to manage.

A questionnaire for speaker candidates drawn up by the Freedom Caucus ahead of a Tuesday night forum, first published last week by Politico, sketched out a series of demands: more rank-and-file representation on the crucial Republican Steering Committee; adherence to the “Hastert rule” requiring a majority of Republicans to support any bill brought to a floor vote; and an end to retaliation for opposing leadership on procedural votes.

The Freedom Caucus has so far thrown its support behind Rep. Daniel Webster (R-Fla.), who has based his candidacy almost entirely on a pledge to “push down the pyramid of power.” He has highlighted his experience overhauling the Florida state House of Representatives as speaker in the mid-1990s, where he eliminated subcommittees and decentralized control of the floor schedule.

“The only way to improve the product is to fix the process,” Webster wrote in a short manifesto, “Widgets, Principles and Republicans,” that has gained currency among the GOP’s right flank.

After hearing Webster speak at the Tuesday forum, Rep. Thomas Massie (R-Ky.) spoke warmly of his proposals — such as abandoning the decades-old practice of considering most significant legislation on the floor under “rules” that limit debate and amendments.

“Before we vote on every bill, we vote to . . . limit our authority as rank-and-file members,” Massie said. “We don’t need rule votes. We have a rule manual.”

Republican leaders point out that those procedures don’t exist to sideline the Thomas Massies of the GOP conference; they exist to sideline the minority party.

Under open rules, Democrats could propose embarrassing amendments calibrated to provide fodder for ads targeting vulnerable Republican incumbents — or they could simply propose enough amendments or raise enough parliamentary points of order to keep a bill on the floor for days, grinding the majority agenda to a halt. The Hastert rule would become a dead letter.

“You couldn’t handle anything sensitive if you just opened it up,” said Rep. Pete Sessions (R-Tex.), the Rules Committee chairman. “For every argument that you want to make that we ought to get away from something, you ought to realize there’s a reason why we’ve got it.”

The demands to tinker with the mechanics of party discipline — particularly calls to rework the Steering Committee, which doles out committee assignments and chairmanships — are more nuanced. They reflect the hard right’s desire for more muscle in the Republican Conference, but they also reflect the remarkable turnover in the GOP ranks.

More than half of the House GOP ranks — 56 percent — has served three terms or fewer, and only one of those members holds any of the committee chairmanships or leadership posts that get ex-officio representation on the Steering Committee.

The hard-liners want more weight given to the rank-and-file on the committee by allowing the speaker one vote instead of the current five, increasing the number of seats allocated on a geographic basis from 13 to 20 and stripping committee chairmen of their ex officio seats.

In another potentially radical reform, committee chairmen would be chosen by secret ballot of the Republican members of that panel — curtailing the Steering Committee’s power to dole out chairmanships as a reward for fundraising, party loyalty or seniority.

Those changes hold great appeal for relative newcomers who pledged to quickly make a difference in Washington.

“We need to change the business model a little bit,” said Rep. Curtis J. Clawson (R-Fla.), a Freedom Caucus member who has been in office for 15 months. “We’re at an inflection point, and we have a chance to make real change in how we govern, how we select folks, and it’s getting by us.”

Changes to the rules hold less appeal for veteran lawmakers who have spent their congressional careers playing by them, such as Rep. Tom Cole (R-Okla.), who has earned several trusted positions during his seven terms, including chairmanship of the National Republican Congressional Committee and the chair of an Appropriations subcommittee.

If “you want to do well around here, you work,” Cole said Friday. “I’ve never found it that hard to work yourself to where you wanted to go in this Congress. But you do have to work, and, guess what, you don’t just show up and get to rule the world. It takes some time.”