Anna Greenberg is senior vice president of Greenberg Quinlan Rosner Research, a Democratic polling firm.

Democratic presidential nominee Hillary Clinton acknowledges the crowd before delivering her acceptance speech at the Democratic National Convention in Philadelphia on July 28. (Melina Mara/The Washington Post)

Hillary Clinton is the most admired woman in the United States.

Hillary Clinton is viewed unfavorably by nearly 60 percent of Americans.

Somehow, both are true at the same time. How can that be?

The answer says a lot about women who break glass ceilings.

No woman has been in the public eye more in the last quarter-century than Clinton — and no woman has been so venerated. According to Gallup’s annual survey, the former first lady, U.S. senator and secretary of state has been rated the most admired woman in America 20 times over the past 23 years, including in its most recent December 2015 survey. Since the question was first asked by Gallup in 1948, Clinton has been top rated more than any other woman, besting even Mother Teresa; only Eleanor Roosevelt comes close, having been rated most-admired for 13 years in the 1940s and 1950s.

Democrats built on this admiration at the Democratic National Convention this week. “No matter how daunting the odds, no matter how much people try to knock her down, she never, ever quits,” President Obama said. “That’s the Hillary I’ve come to admire.” First lady Michelle Obama hit the same theme: “What I admire most about Hillary is that she never buckles under pressure.” Speaker after speaker reminded the hall that she’s been taking on battles since her early 20s, always in the service of others and always leading higher levels of achievement.

And yet, over the years, a significant majority of Americans has held unfavorable views of Clinton. Her overall favorability rating is historically low for the nominee of a major party (as is Donald Trump’s), but the negativity extends to perceptions of her character as well. In the 1990s and today, questions about her honesty and trustworthiness have been her biggest vulnerabilities.

Various factors contribute to this disconnection between admiration of Clinton and affection for her, but the most convincing explanation is her singular role as a woman at the highest level of American government.

Looking at the fluctuations in Clinton’s favorability over time, it seems clear that she suffers the backlash that occurs when a woman breaks gender-role barriers. When she follows traditional roles — say, being the supportive wife or loyally serving President Obama as secretary of state after she lost to him in 2008 — her favorability rises. When she defies traditional gender roles — taking on health-care reform as first lady, running for Senate and then president — her favorability declines. People admire her for shattering tradition — but they don’t necessarily like her for it.

Women who break barriers are rarely popular; to quote feminist historian Laura Thatcher Ulrich, “well-behaved women seldom make history.” Even today, people are ambivalent about feminism and feminists. But ultimately, the women who break these barriers are admired for the same things people like about Clinton: their grit, tenacity and, indeed, their willingness to be unpopular and unconventional.

In focus groups I conduct across the country, many swing voters are ambivalent about Clinton. Voters talk about her resilience and tenacity; they know that she has experienced trials and tribulations unlike any other woman in public life, and yet she not only lands on her feet but excels in her new professional challenges. Older women, especially, see a woman who managed to keep her family intact under hideous personal scrutiny, while making an amazing transition from first lady to secretary of state and now the first woman to be nominated for president by a major party.

At the same time, they feel as if they do not know her deeply and they wonder about her motives; they do not know what makes her tick because many do not connect with a woman who defies norms in the way she has. Clinton herself acknowledged on Thursday night, “I get that some people just don’t know what to make of me.” Indeed, in research for other female candidates, I’ve heard some people express skepticism about women who have made different life choices than they have, particularly pursuing career over family. They just do not relate to a “woman in a suit with a plan.”

In this context, Clinton Foundation donations and private email servers compound the notion that she is inscrutable and untrustworthy. This anecdotal evidence is supported by polling, which shows decisively that her experience and steadiness are her most important attributes and the most important contrast with Trump, while honesty and trustworthiness are her biggest vulnerabilities.

Consider Clinton’s history and note the association of her life choices with her favorability over time. In Arkansas, she became a successful partner in a law firm and did not change her last name when she was married; ultimately she took the Clinton name to help her husband regain the gubernatorial seat he lost. At the start of Bill Clinton’s 1992 presidential campaign, when he suggested voters would get “two for one” if they elected him, only 38 percent had favorable feelings towards her. She was criticized for having a career when Bill Clinton ran for president and sparked controversy by saying “I suppose I could have stayed home, baked cookies.” She rebounded in popularity — to 56 percent — after she defended her marriage on “60 Minutes” prior to the New Hampshire primary, which most credit with reviving Bill Clinton’s campaign.

Clinton carved out a different role as first lady taking on policy issues like health-care reform and championing women’s rights, most notably in her 1995 speech in Beijing. After the inauguration, her ratings were as high as 67 percent favorable, but took a hit – to 48 percent — when she spearheaded the failed health-care reform effort. Again, Clinton’s standing rebounded in 1998 to as high as 67 percent when she defended her husband in the wake of the Monica Lewinsky scandal.

Clinton’s favorability suffered — dropping to 45 percent — when she ran for the Senate, when people suggested she was an unqualified carpetbagger. She came out of the 2008 primary damaged, in part, by attacks that she did not know how to “run as a woman,” endlessly scrutinized on her looks and emotions. But her favorability soared —to 66 percent — when she served as secretary of state, in part because of her unbelievable stamina and unrelenting work in restoring the United States’ standing in the world, but also because she served as a loyal Obama soldier after her loss in the primary.

In the end, the most admired women are the ones who challenge gender norms, even if it is unpopular and makes them unpopular. Eleanor Roosevelt was chastised in her day for her women-only news conferences, her column that delved into domestic and political issues alike and her public advocacy on policies not even supported by her husband. She was criticized for overreach from the traditional role first ladies play and for her appearance, which was deemed insufficiently feminine. And yet she is just behind Clinton as the most admired woman in the United States.

If Clinton wins in November, it’s a sure bet that she’ll be the most admired woman by a large margin. Whether Americans come to have a favorable view of her may say less about her than about us, and our ability to accept her as a gender-role iconoclast.