Late one night in January, 2005, Carly Fiorina sat in a hotel room in Davos, Switzerland, where she was attending the World Economic Forum, in a state of angry dismay. She was the C.E.O. of Hewlett-Packard, and she believed that, in an effort to undermine her, members of the board were leaking confidential information about the company to the press. She had instructed lawyers to question all the members, so that they could “come clean.” Now, on a conference call, they still denied the leaking. Two weeks later, the board fired her. As she writes in “Rising to the Challenge,” her latest memoir, “Fearing for their positions, they behaved in an unprincipled fashion and ousted me from mine.”

Others have portrayed events differently, attributing Fiorina’s termination to unhappiness over H.P.’s merger with Compaq. She had sold the deal brilliantly, amid a bitter proxy fight, but the execution was badly managed, and the value of the stock fell. All this might have been of interest solely to business-school case writers had not Fiorina unexpectedly risen to the top tier of Republican Presidential contenders, joining the two other non-politicians in the race, Donald Trump and Ben Carson. In G.O.P. circles, she is being greeted as a savior of, if not the Party’s electoral prospects, its sanity—as someone who might allow the focus to return to candidates like Jeb Bush. It’s an awkward space, at the intersection of outsiderdom and fear of the Donald, but, for the moment, Fiorina has claimed it.

“It’s only in this country that you can go from being a secretary to the chief executive of the largest technology company in the world,” she told Jimmy Fallon, on “The Tonight Show,” last week. “Wow,” Fallon said. “It’s unbelievable.” It’s also, as with much that Fiorina says, a little more complicated than that. In 1976, Cara Carleton Sneed graduated from Stanford, where her father had been a law professor. (He later served in the Nixon Administration and on the Ninth Circuit Court of Appeals.) She enrolled in law school at U.C.L.A. but dropped out. She worked briefly in a real-estate firm as a receptionist, then got married, moved to Italy for a while, and returned to attend business school in Maryland, after which, Stanford degree and M.B.A. in hand, she was hired as a management trainee at A. T. & T. That was where she met her second husband, Frank Fiorina, a mid-level executive.

She quickly became known as a sales prodigy, a reputation that grew when she became a division chief at Lucent, a company formed from A. T. & T.’s telecommunications-equipment business. Some of the deals she closed were, in fact, unbelievable. In 1999, Lucent said that a little-known firm called PathNet would buy as much as two billion dollars’ worth of its equipment. As Fortune noted later, PathNet’s annual revenues were only $1.6 million; Lucent would loan it money for the sale, which was unlikely to be repaid. But by the time such dubious accounting became public, leading to a collapse of Lucent’s stock, Fiorina, who was never accused of wrongdoing, had left for H.P., with a signing bonus worth sixty-eight million dollars and millions more in pay. When H.P. fired her, she got a twenty-million-dollar severance package, plus fifteen thousand dollars for career counselling. Only in this country, perhaps, could a C.E.O. receive compensation worth more than a hundred million dollars in six years, get fired, and use the money to enter politics.

Fiorina’s first run for office, in the 2010 U.S. Senate race in California, is best remembered for an Internet ad, produced by her campaign, that portrayed her primary opponent as a demon sheep with glowing red eyes. She won the nomination, then, in the general election, produced a video showing the Democratic incumbent, Barbara Boxer, as a swollen disembodied head. In the current primary campaign, Fiorina has been the target of some misogyny, particularly from Trump, so it is instructive to contrast her dignified response to his comments about her face and her voice—“I think women all over this country heard very clearly what Mr. Trump said”—to the use, in her anti-Boxer ad, of footage of fingernails scraping a chalkboard. (Boxer responded with an ad noting that Fiorina had fired thirty thousand employees at H.P., and she won by a million votes.) It is a contradiction that Fiorina seems to revel in; in her memoir, she decries sexism and, in the next paragraph, rejects the “feminist movement” as “politicized” and “captured by a left-wing agenda.”

For Fiorina, the center of that agenda is reproductive rights, or, as she puts it, the “butchery” of abortion. In the most recent G.O.P. debate, she called for the defunding of Planned Parenthood and dared President Obama to look at an undercover video about the organization, made by an anti-abortion group: “Watch a fully formed fetus on the table, its heart beating, its legs kicking, while someone says, ‘We have to keep it alive to harvest its brain.’ ” There are graphic segments in the video, but it does not show what Fiorina described. Yet, when Chris Wallace, of Fox News, asked her to acknowledge that “there is no actual footage of the incident,” Fiorina replied, “No, I don’t accept that.”

As a campaigner, she is more adept than Trump at pivoting from blunt personal attacks to confident policy proposals that tend to be described as “crisp”—full of details, if not facts. Of the Affordable Care Act, she says, “What you see is emergency-room visits are up over fifty per cent.” (There is no apparent source for this number.) As an executive, she can “read the fine print” of the science on climate change, which, according to her, shows that there’s nothing the U.S. can do about it. California’s drought, she contends, is a “man-made disaster,” not because of climate change but because “liberal” politicians, worried about fish, prevented the state from building dams. “That’s pretty dumb,” she told Chuck Todd. (One of those conservation-minded politicians was Ronald Reagan.) Businesspeople, she says, are “accountable.” Politicians are not.

And yet Fiorina defends her record at H.P. with numbers that are often muddled (pointing to revenues instead of profits) or opaque (“We tripled innovation”). If Trump’s success is a testament to brash celebrity, Fiorina’s signals the power of content-blind marketing, at least in the short term. In recent years, H.P. has gone through so many changes that it’s hard to tease out what her effect on its long-term prospects has been. (Meg Whitman, the current C.E.O., recently announced that the firm would split, cutting up to thirty thousand jobs.) In the unsettled period after Fiorina’s departure, Fortune reported, employees sometimes ignored managers’ instructions, a maneuver they called “flipping the bozo bit.” It sounds like a G.O.P. code name for a Trump-removal operation. If Fiorina closes that deal, what’s next? ♦