TWITTER erupted on Wednesday, as Twitter likes to do, over a story about three topics—sexism, women in leadership and the New York Times—that, when combined, are a potent cocktail for social media.

An anonymously sourced story in Politico about the “brusque” and “stubborn” leadership style of the Times’ executive editor, Jill Abramson, was widely denounced for being unfair and, some said, sexist in its reporting. Critics questioned whether it would still be considered news if if the “difficult” or “condescending” figurehead had been a man. Others argue that men at the top of the Grey Lady—or any number of other newsrooms, for that matter—have been trashed for such behaviors before. (In a defense of his piece, the author makes a similar argument.)

This would all be little more than journalistic navel-gazing if the discussion surrounding the piece weren’t so important. Whether or not the Politico story was sexist, the fact remains that female leaders are indeed judged twice, both by how well they actually perform and by how well they fit in with how we think women should behave.

Don't buy it? Writer Farhad Manjoo pointed out the discrepancy on Twitter (where else?) by comparing two eerily similar anecdotes. The Politico piece recounts a story of how Abramson, unhappy with a photo on the paper's homepage, reportedly told an editor in a meeting: "I don’t know why you’re still here. If I were you, I would leave now and change the photo.” It's designed to be an illustration of her bluntness. But in a Fortune profile of Apple CEO Tim Cook, a similar episode is recalled in order to describe the tech leader as an unflinching operations whiz. A problem in Asia came up during a meeting, and thirty minutes later, Cook reportedly asked his lieutenant "why are you still here?" The executive immediately headed to the airport and booked a flight to China, a story that is "vintage Cook: demanding and unemotional."

This is a well-documented phenomenon. Catalyst, a research organization that studies women in leadership, calls this the “damned if you do, doomed if you don’t” dilemma. Their research shows that those who judge female leaders are a little like Goldilocks: Women in power are always too hard or too soft. They can be capable or liked, but not both.

In her now ubiquitous book, "Lean In," Facebook COO Sheryl Sandberg devotes an entire chapter to the “success and likeability” conundrum, in which women who get promoted don’t necessarily become more liked. She quotes the frequently cited Heidi/Howard case study—research showing that when the same highly successful leader is described to grad students, she’s seen as far more appealing when she’s given a male name instead of a female one. When women violate the stereotypes we have of them as nurturing, sensitive and communal, they’re perceived more negatively.

Our response to these biases is so entrenched that Sandberg doesn’t even advocate trying to change them. Rather, she suggests more culturally acceptable ways for women to negotiate (“whenever possible, women should substitute ‘we’ for ‘I’”) and recommends combining “niceness with insistence,” a style one university president calls “relentlessly pleasant.” Sandberg is a bit sheepish—“I understand the paradox of advising women to change the world by adhering to biased rules and expectations”—but writes that further change won’t come until more women are in power.

This is why all the chatter around the Abramson Politico piece does matter outside of the media fishbowl. From the sound of it, Abramson may not be relentlessly pleasant or insistently nice. And she shouldn’t have to be. But until we get enough women into positions like hers, they are either going to need to shine a spotlight on the double bind they’re in or end up continuing to play by those biased rules.

Jena McGregor is a columnist for On Leadership.

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