Independents are a finicky and fickle bunch. They are deeply dissatisfied with the direction of the country, with a majority saying things are badly off track. Social issues do not concern them. Many voted for President Obama.

And now, this critical voting bloc — which makes up roughly 45 percent of the New Hampshire electorate — is about to take center stage. If the Iowa contest exposed the conflicted mood and sharp divisions within the GOP, then New Hampshire will offer a snapshot of a hard-to-pin-down sliver of the electorate that has an outsize impact on elections, particularly in this swing state.

“I’m disappointed in the unemployment rate and the economy, so I’m open and I’m listening,” said Julie Gagne, 52, an independent who voted for Obama in 2008.

Now, Gagne is planning to vote in Tuesday’s Republican primary. As part of her decision making, she went to see Mitt Romney last week at an event outside a chowder shop.

A Republican-leaning voter who broke from the party only to back Obama, Gagne soured on the president last year after losing her job. She said she was upset to see Obama focus on health care instead of the economy. Gagne went to see Romney in hopes of connecting with the man she believes will probably be the Republican nominee.

But to her dismay, Gagne found Romney “choreographed and deliberate and stiff and staged,” she said. She most likely will vote for Romney in the primary but reserve judgment about the general election. “I guess I’ll just keep an eye on the economy and just watch how the campaign plays out,” she said.

That sentiment is reverberating across the country as independents express their dissatisfaction with both the president and the alternatives. Their unhappiness could not matter more: Nationally, the candidates will be competing for the roughly one-third of the electorate that is considered independent and remains up for grabs.

It is not clear how Obama and his eventual Republican opponent will play with this exceptionally disgruntled group of voters.

Independent voters — more than eight in 10 of whom are displeased with the country’s political system — are more deeply unhappy with Washington than are Democrats or Republicans, and they’re far more apt to blame both sides in Congress.

Fully 39 percent of independent voters in Washington Post-ABC News polling said they’re “angry” about the way the federal government works, higher than the numbers among Democrats and Republicans.

They are no less critical of the candidates. While Obama appears to have disappointed many independents who supported him in 2008, Romney has hardly locked them down. Even though Romney holds a double-digit lead in recent New Hampshire polls, his support is weaker among independents, who have also flocked to Rep. Ron Paul (Tex.) and former Utah governor Jon Huntsman Jr.

“Romney’s efforts to reassure the right wing have made him suspect here” among independents, said Linda Fowler, a Dartmouth College government professor. Paul, she said, has flourished because of his libertarian bent and antiwar views.

In part, independents are on display because of New Hampshire’s voting rules. Unlike in Iowa, where voters had to declare their allegiance to the Republican Party to participate in the caucuses, unaffiliated New Hampshire voters are permitted to vote in either party’s primary. Without a competitive Democratic primary, even Democratic-leaning independents might be inclined to cast ballots for Republicans, regardless of how they plan to vote in the general election.

Most participants in the Republican primary will be true Republicans or Republican-leaning independents, said Andy Smith, a political science professor at the University of New Hampshire. But about 10 percent will be independents or Democratic-leaning independents, offering a glimpse of how an important swing group is shaping up. (Registered Democrats are not permitted to vote in the GOP primary in New Hampshire.)

At a town hall meeting hosted by Huntsman last week, several independents who backed Obama in 2008 said they were exploring their options. Even Obama’s supporters expressed interest in the Republican primary, saying they were unsure that the president will be able to win this year.

“I think it is probably 50-50 that the next president is a Republican, so I think it’s very important to get it right,” said John Nelson, 49, an Obama supporter.

Karen Cahillane, a psychotherapist from Concord, said she voted for Obama in 2008 and plans to again. “But I’m listening,” Cahillane, 58, said.

A cancer survivor, Cahillane said a top priority for her is health care. She said she has been disappointed in Obama’s ability to push his agenda. “I read his book Audacity of Hope,’ and I thought, ‘He gets it,’ ” she said. “I think every president’s hands are tied more than they think they are when they are up here stumping.” Her weariness extends beyond Obama, however, and although she is open to voting for a Republican, she says she probably will back Obama again.

Ginny Tirrell, 53, a seamstress at Globe Manufacturing, where Huntsman made a campaign stop Wednesday, said that four years ago, she voted for Obama — “unfortunately.” This year, she is planning to vote in the Republican primary and is deciding between Huntsman and Paul. But Romney is not on her radar screen, she said. “Something about him just turns me off,” she said.

The candidates have been attracting more politically mixed crowds than they did in Iowa. Among the voters at a Newt Gingrich town hall meeting Wednesday in the New Hampshire town of Laconia were a number of independents, as well as some registered Republicans who voted for Obama.

Retiree Kim Trask, 57, is unaffiliated but has voted Republican since the Sept. 11, 2001, attacks. She is considering backing Gingrich or former Pennsylvania senator Rick Santorum — but not Romney. “I think he’s a little bit too moderate for me,” she said.

A few rows back sat retirees Bob and Judy Chase. He’s a Republican; she’s a Democrat. In a show of their political independence, however, Judy wore an elephant in one ear and a donkey in the other.

Both are intrigued by the Republican candidates now descending on their state, but both approve of Obama’s leadership. “He was a little naive, but I think he’s done a fairly good job,” Bob Chase said.

Judy Chase described herself as supportive of same-sex marriage and abortion rights but said she is troubled by the tendency of some people to vote exclusively on those matters. So she and her husband have another candidate to check out: Santorum.

Polling analysts Scott Clement and Peyton Craighill contributed to this report.