WASHINGTON — The paradox of Vice President Mike Pence getting in trouble for failing to wear a mask is that perhaps no member of the Trump administration is more shrouded behind an invisible one of his own making.

For weeks now, he has stood day after day at the side of President Trump, stone-faced and unreadable, never displaying a hint of reaction to the president’s governor-bashing tantrums and bleach-injecting prescriptions, while offering a calming, measured, what-he-meant-to-say counterpoint along with constant flattery for the boss.

In the most consequential mission of his career, Mr. Pence has tried to navigate the complexities of a mysterious disease and the vagaries of a mercurial president at the same time, steering the response to the most deadly pandemic in generations without getting caught up in the melodrama of the moment. Yet questions have lingered about how seriously he himself took the threat at first and what advice he gave the president in the days when it really mattered.

By touring the Mayo Clinic this past week bare faced while everyone else around him was masked in keeping with the medical center’s policy, Mr. Pence generated a sharp backlash for not adhering to the very precautions he himself has advanced. By showing up at a ventilator plant two days later with a mask covering his face, he also showed that, unlike the president, he was willing to back down in the face of criticism. But then a reporter said his aides threatened retaliation for what they claimed was a violation of an off-the-record agreement.

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Pence Tours Mayo Clinic Without a Mask

Vice President Mike Pence did not wear a mask while visiting coronavirus patients at the Mayo Clinic in Minnesota, a violation of the facility’s policy.

“Hi doctor.” “Vice President Pence, nice to meet you.” “Great to meet you too. Great job. You went through the coronavirus?” “Yes my wife and I both had it.” [inaudible] [inaudible] “The president I often talk about a whole of America approach. You’ve got the whole of Minnesota approach.”

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Vice President Mike Pence did not wear a mask while visiting coronavirus patients at the Mayo Clinic in Minnesota, a violation of the facility’s policy.CreditCredit…Nicholas Pfosi/Reuters

The controversy over the mask was a rare miscue for a vice president who has stuck to the script and drawn praise both from Democrats who prefer working with him over the president and from Republicans who privately wish he were the one in the Oval Office. In the good-cop-bad-cop tandem, Mr. Pence takes the calls from the governors seeking help and even calls them unbidden to ask what else he can do.

“This has been a real defining moment for him,” said Gov. Larry Hogan of Maryland, a Republican and chairman of the National Governors Association who has been at odds with the president over the response to the coronavirus but like other governors has turned to Mr. Pence to work through problems. “In all honesty, I think he has risen to the occasion. I think he’s done an amazing job.”

At the same time, Mr. Pence has been an enabler, never contradicting the president’s many falsehoods and advancing some of his own about the administration’s handling of the crisis. His nonstop praise for “the president’s leadership” and “decisive action” risks sounding unctuous. And he has not been able to keep Mr. Trump fully committed to policies recommended by public health experts to curb a virus that has already killed more than 65,000 people in the United States.

“Behind the scenes, whatever he is saying, if it is different from what he is saying in front of the cameras, is not working at all,” said Kathleen Sebelius, a former secretary of health and human services under President Barack Obama. “And in front of the cameras, he is playing a role as a sidekick to a president who is delivering, at the very best, very mixed messages to the public which could put them at risk.”

Critics inside the administration, who asked not to be identified, acknowledge that time was squandered during the weeks before the president began to embrace quarantine policies. They said they were mystified Mr. Pence had escaped blame for the slow response.

As with much of the relationship between the president and the vice president, it has been difficult to discern what goes on between them. “I think the vice president has always viewed his role as making sure the president has the best information possible to make decisions for the American people,” said Marc Short, Mr. Pence’s chief of staff. Aides to Mr. Pence said that included conveying the advice of the medical experts.

Mr. Trump gave Mr. Pence the unenviable assignment of taking over the administration’s coronavirus response in late February, a decision seen as fraught with danger for a vice president some thought was being set up as a fall guy if things went badly. Mr. Trump privately described the role to advisers as short term and played down its significance.

Once Mr. Pence took over, administration officials said task force meetings focused more on spin rather than policy, going from operational and decision-making discussions run by Alex M. Azar II, the secretary of health and human services, to mostly prep sessions for the news briefings that the vice president initially led until Mr. Trump decided he should do them himself. Some officials complained that Mr. Pence’s office abruptly ended a daily morning conference call that Mr. Azar’s team had been leading.

Mr. Pence’s aides took over communications surrounding the virus, insisting they approve interviews, a move that prompted concern that the vice president was trying to silence experts. Mr. Pence’s staff went into overdrive to dispute that, making Dr. Anthony S. Fauci, the federal government’s top infectious disease expert, available for interviews with different outlets. Eventually, Dr. Fauci began booking interviews himself, aggravating aides to both the president and the vice president.

While complimentary of Mr. Pence himself, some administration officials complained that his staff was less concerned about issues like ramping up testing, manufacturing ventilators and repatriating American citizens stuck overseas. Mr. Short has made clear to colleagues that he thinks the lockdowns have gone too far, although the vice president is said not to share that view.

White House officials disputed that Mr. Pence’s team did not focus on issues like supplies and testing early on and said he took over a task force that was dysfunctional under Mr. Azar. They point to a series of actions the administration took in the weeks after Mr. Pence’s takeover on Feb. 26, including suspending travel from Iran, issuing a global travel advisory, increasing inspections of nursing homes, setting up an emergency operations center and pushing to improve testing.

Mr. Azar skirted around any suggestion of tension in a statement on Saturday, saying he was “proud to work” with Mr. Pence and calling the vice president “instrumental not only in marshaling resources from across the federal government but also coordinating with state governments and countless American companies and nonprofits.”

At the start, Mr. Pence’s own steady, mature, relentlessly polite public appearances won wide praise as he conveyed the seriousness of the virus, expressed concern about the victims and projected resolve to combat the pandemic.

“Pence is really good at this. Don’t @ me,” Jane Lynch, the actress and frequent Trump critic, wrote on Twitter on March 10, leading many liberals to express their disagreement anyway. Michael Gerson, a former senior adviser to President George W. Bush, wrote in his Washington Post column, “Never has the phrase ‘President Pence’ had a better ring to it.”

Recognition for Mr. Pence, of course, was bound to upset Mr. Trump, who likes to be the center of attention, and he quickly took over the briefings. At one point, he made a point of putting the vice president in his place, mocking Mr. Pence’s artful dodge of a question about the administration’s refusal to reopen enrollment in Mr. Obama’s health care program. “I think that’s one of the greatest answers I’ve ever heard because Mike was able to speak for five minutes and not even touch your question,” Mr. Trump said.

Still, Mr. Pence seems to know when to heed Mr. Trump and when to quietly ignore criticism of him. Whenever the president rails against governors who have criticized him, the vice president simply remains silent. At one point in March, Mr. Trump admitted that he sought to cut off Democratic governors who were not “appreciative” enough of him.

“I say, ‘Mike, don’t call the governor of Washington. You’re wasting your time with him. Don’t call the woman in Michigan,’” Mr. Trump said.

In interviews, both Govs. Jay Inslee of Washington State and Gretchen Whitmer of Michigan said Mr. Pence never stopped calling and never endorsed the president’s vitriol, but simply proceeded as if it had not happened.

“The vice president has a good poker face,” said Mr. Inslee, who was called a “snake” by Mr. Trump even as he was treated professionally by the vice president. “I would hope he’s frustrated by it and I don’t press him on that because he’s in a difficult position. But I try to provide him encouragement to be as effective as possible to reduce the unnecessary tensions that are caused by the president’s eruptions. He mostly nods his head, understands our position and doesn’t betray his inner thinking.”

Ms. Whitmer said she has always found Mr. Pence “very accessible and cordial” regardless of what Mr. Trump said. “I’ve not had a hard time getting in touch with the vice president,” she said. “He will on occasion call totally out of the blue unscheduled just to check in or to tell me about something coming from” the Federal Emergency Management Agency.

Another governor, who asked not to be named, said Mr. Pence never betrays what he really thinks about the president’s behavior. “Multiple times, I say, ‘But Mike, this is just crazy,’ and sometimes there will be a little pause and I’m waiting for him to say, ‘Yes, I know.’” But he never does. Instead, Mr. Pence says, “I understand. Thank you for your input.”

Senator Brian Schatz, Democrat of Hawaii, who has attended briefings with Mr. Pence, said the vice president remains disciplined no matter how provocative the question. “He is a skilled communicator and is very good at absorbing criticism,” Mr. Schatz said. “Almost every answer to every question begins with an acknowledgment of how important the question is and maybe some praise for your home state governor. At that level, he’s world class.”

“In terms of answering the question in a substantive way,” Mr. Schatz added, “I think he’s stuck because the administration has failed and there’s nothing he can do in private or in public to spin that.”

Mr. Pence has been pressed to reconcile his own initial statements with the reality that followed. In early March, he promised that the government would distribute 4 million test kits within a week. But it was not until mid-April that the nation actually conducted 4 million tests.

The vice president’s response was to make a distinction between distributing tests and actually conducting them. “I appreciate the question,” he told a reporter who asked about it this past week, “but it represents a misunderstanding on your part and, frankly, a lot of the people in the public’s part about the difference between having a test versus the ability to actually process the test.”

In the end, the crisis has both put Mr. Pence on a much more visible stage than ever while simultaneously demonstrating the limits of his influence in an administration headed by Mr. Trump. Tim Pawlenty, the former Republican governor of Minnesota, said people expecting Mr. Pence to force the administration one direction or another were misguided.

“Task forces and vice presidents don’t get to decide things,” he said. “They just get the opportunity to nudge the boss in a certain direction — and the V.P. has been a good nudger under difficult circumstances.”

Peter Baker reported from Washington, and Maggie Haberman from New York.