PARIS — For at least the last three French elections, voters in the town of Louviers, about 60 miles northwest of Paris, have cast their ballots for the candidate who ultimately won the presidency. So who are they going to vote for Sunday, in the country’s closest race in memory?

“I haven’t decided. It’s gnawing at me,’’ said Charlène Hedoux, 30, a cleaning woman who was sitting at a bus stop last week in central Louviers, which has a soaring Gothic church and bustling cafes. “I have children. I didn’t before. When one sees how these last few days have been going, it’s not very reassuring.’’

And that was before the terrorist attack Thursday that left a police officer dead in central Paris and added yet another combustible element to an already volatile race.

The election Sunday is one of the most consequential in recent times — not just for France, but for Europe — and one of the most unpredictable, too.

Even at this late stage, a remarkable 28 percent or so of voters remained undecided. The four leading candidates span the extremes of the political spectrum and are locked in a virtual dead heat.

The potential outcomes are just as broad. Depending on who wins, France could seek to leave the European Union and recast security alliances with a tilt toward Russia.

Some would have France take harsher stands on Muslims, immigration, and domestic security.

Alternately, France may elect a president who wants to shrink the ranks of civil servants, eliminate some job protections, and reduce France’s generous welfare state, making the country more competitive in the global economy but risking a popular backlash. Yet another possibility: a winner who aspires to sharply increase taxes on the rich and nationalize banks.

The two candidates who receive the most votes Sunday will compete in a runoff May 7.

“There’s never been a campaign where the uncertainty was so uncertain,’’ said Edouard Lecerf, global director for political and opinion research for Kantar Public, a public opinion research firm.

“The mistrust of politicians is stronger than it has ever been.’’

That mistrust appears to have caused many voters to veer away from traditional politicians in search of someone they feel is more principled, prompting even candidates who are insiders to claim they are “outsiders.’’

A majority of French voters have traditionally supported parties with established ideologies, either mainstream left or right. But traditional left-right allegiances are breaking down all over — in Europe, as they appear to have in the United States — as polarization grows.

Globalization is a stark dividing line, with candidates on both the extreme left and right crusading against it, and more centrist candidates embracing it.

Voters are looking to get away from politics as usual. That appears to have helped one of the leading candidates, Emmanuel Macron, a former economy minister who has formed a new party with a platform that combines pro-business, pro-European Union, and pro-social welfare elements — a novelty in contemporary France.

With allegiances to existing parties diminishing, some French voters find themselves torn between candidates who are diametrically opposed.

A good example was Pierre Haux, a teacher at a technical school, who went to a rally last week in Lille of the mainstream conservative candidate, François Fillon, who is under the cloud of a nepotism scandal that has led to claims of embezzlement.

Haux said he was also weighing voting for the far-left candidate Jean-Luc Mélenchon, but worried that he was “a one-man show.’’

In France, as elsewhere, the election may be decided in the provinces, in smaller and midsize towns and rural communities. Rural areas have proved especially unpredictable in recent votes, like Britain’s so-called Brexit vote to leave the European Union and Donald Trump’s election in the United States.

In addition, the trend lines — whether a candidate appears to be steadily gaining votes, losing them, or remaining flat — are of little help. Most recently they have shown the third- and fourth-place candidates, Fillon and Mélenchon, gaining ground, while the two in the lead, Macron followed closely by the far-right candidate Marine Le Pen, have remained flat or lost ground.

For French voters, who are facing 10 percent unemployment, with levels closer to 25 percent for young people, the most pressing issues are economic, but even those can lead in multiple directions.

Some voters attribute France’s economic ills to poor leadership, others to globalization, and still others to an influx of immigrants into the country.

Unlike in other years, nothing in the current election has followed old patterns.

It is a cliché of French politics that voters follow their hearts in the first round, choosing the person they most want, but follow their heads in the runoff, choosing one less likely to do damage.

That equation may no longer apply.

Because the race is so close, voters may try to vote strategically from the beginning and not risk an outcome in which two extremist candidates reach the second round.

On the other hand, if tradition holds true, and voters do follow their hearts, it is not impossible to foresee a final round where voters must choose between two candidates at the extremes, Le Pen on the right and Mélenchon on the left.

Significant numbers of voters also appear likely to abstain or to cast a blank piece of paper to show their dissatisfaction with all of their choices. That, too, would seem to favor the nonestablishment candidates, whose supporters are more motivated to go to the polls.