It may have seemed like made-for-TV farce: Two people showed up at a government agency this week, each claiming to be the interim director.

But underlying that Washington drama this week at the Consumer Financial Protection Bureau (CFPB) is a serious question over how independent regulatory bodies should function.

Without enough independence, an agency can’t regulate industry effectively. Without enough political accountability, there are few checks and balances to ensure it is carrying out the will of voters. Resolving that tension is difficult.

“The short answer is: It’s inevitable,” says Jim Doig, a research professor of government at Dartmouth College in Hanover, N.H.

There are exceptions. Sometimes, independent agencies can operate with little political input, especially when scientific or professional expertise is paramount, such as the New York Port Authority deciding on where to put bridges to ease traffic flow.

But at the CFPB the question is now front and center.

The interim director appointed by President Trump – Mick Mulvaney, director of the Office of Management and Budget – arrived at the agency Monday toting doughnuts, and issued a 30-day hold on new agency actions and hires. Deputy Director Leandra English, who by law was to serve as interim director when Director Richard Cordray left the job, sent out an email welcoming employees back to work then went to Capitol Hill to consult with key Democratic supporters of the agency.

By Tuesday, a federal judge has ruled in favor of the president’s choice. So, for the moment, Mr. Mulvaney is interim director while Ms. English is deputy director. But the matter appears far from settled and awaits further legal decisions and appeals,

The confrontation revolves around how independent the CFPB should be.

In the aftermath of the 2008-09 financial crisis, the Obama administration and Democrat-controlled Congress were eager to find ways to protect consumers from financial institutions’ misdeeds.

“The track record of the financial regulators leading up to the crisis was exceedingly bad,” recalls Michael Barr, then an assistant secretary of the Treasury and key architect of the Dodd-Frank Act of 2010, who now teaches law at the University of Michigan. Even the Federal Reserve, which has enjoyed a long history of insulation from politics, missed the danger signs that preceded the financial crisis.

J. Scott Applewhite/AP Leandra English (left), who was elevated to interim director of the Consumer Financial Protection Bureau by its outgoing director, meets in Washington on Nov. 27 with Senate minority leader Chuck Schumer (D) of New York and Sen. Elizabeth Warren (D) of Massachusetts. They discussed the fight for control of the consumer watchdog, after President Trump chose White House budget director Mick Mulvaney for the same post.

So Democrats combined elements that have insulated other agencies and created something unique, reasoning the agency needed protection from pressure from both the industry and future conservative Congresses or administrations that might weaken it.

Among its accomplishments, the CFPB has set stricter standards for the mortgage market and simplified loan disclosures for borrowers. It has eliminated some of the most predatory payday loans and created a database where consumers can publish complaints about financial institutions.

Regulation-wary conservatives have balked at several of these actions. For example, they point out that the complaints in the database are unverified. However, online retailers face the same scrutiny without their sales drying up.

The conservative Competitive Enterprise Institute claims the agency actually hurts consumers, because its regulations have raised the costs of loans and caused banks to stop offering profitable but risky loans.

But that’s a hard case to make against an agency that has forced banks to pay out almost $12 billion to 29 million customers for wrongful practices. Even with the increased costs, banks are making record profits. And those loans that are no longer available include abusive payday loans and the risky mortgages that got so many homeowners in trouble during the Great Recession.

Conservatives’ strongest attack against the agency is that it’s not politically accountable to the three branches of government. Congress doesn’t fund it, the Federal Reserve does. It’s run by a single director instead of commissioners from both parties, which is a more standard setup for independent agencies. And that director, while appointed by the president for a five-year term, can’t be fired by the president, unless there’s “cause,” such as malfeasance or dereliction of duty. (The legal battle over interim directors arose after Director Cordray resigned.) A federal appeals court is due to rule on the constitutionality of the restrictions against a director being fired.

Finally, the Dodd-Frank legislation, which set up the CFPB, instructs the courts to review some of the agencies’ decisions with more than the usual amount of deference.

Some Republicans in Congress have called it a rogue agency. Democrats “created a modern-day emperor, somebody immune from congressional oversight and appropriations process,” Sen. John Cornyn (R) of Texas said in a statement Monday as the drama over dueling interim directors was playing out.

It certainly “is a new agency like no other,” wrote Eric Pearson, an emeritus law professor at Creighton University, in a 2015 essay.

Mulvaney has promised that the CFPB will operate differently in the future, which suggests fewer regulations and a less aggressive stance against financial companies. The immediate question is how long he stays in place.

The legal ruling that favored him over English came from a Trump-appointed judge. But on appeal, in front of a mix of liberal and conservative judges, English’s case has legal merit, says Mr. Doig. Because of the insulation that Democrats gave CFPB from its inception, the decision “would be close.”

That question becomes moot if and when Mr. Trump appoints a new director, who would have to be approved by the Senate. That director would be able to take the agency in a new direction.

A new structure and balance between accountability and independence will depend on Congress.

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“Congress could design this agency in lots of different ways,” says Mr. Barr, the former Treasury official. The CFPB will probably survive, he says.

“Is there a potential for a significant weakening of the agency? I am concerned,” he adds.