California Republicans believe a ballot measure to repeal a gas tax hike offers a golden chance for the party to change its short-term fortunes in a state known as much for its Democratic dominance as its high taxes. Legislators enacted the 12-cent-a-gallon increase last year to remedy a $130 billion backlog in road and bridge repairs. Advocates of Proposition 6 criticize the tax as an excessive burden on lower-income residents, and with support strong among conservatives and independent voters, Republicans hope the party can thwart a “blue wave” on Election Day. “The state’s voters are not as far left-of-center as people might imagine, and the gas tax is an issue that aligns with working-class values,” says Jason Roe, a Republican strategist. The battle pits progressive ideals against pocketbook politics as Gov. Jerry Brown, his political career nearing its end, fights to secure his legacy. “If the tax is going to survive,” says Darry Sragow, a former Democratic strategist, “the governor and his supporters have to start sending a very loud and credible message that it fills a critical need.”

If the categories of threatened species applied to politics, the Republican Party in California would be listed as critically endangered. Chin-tugging researchers would warn that the group appears destined for extinction, offering data that illustrate its depleted condition in the country’s most populous state.

The number of voters registered as Republican has fallen to third behind Democrats and those with “no party preference.” Hillary Clinton tallied 4.2 million more votes statewide than Donald Trump in the 2016 presidential election. Democrats hold almost three-fourths of California’s 53 seats in Congress and both US Senate seats, along with two-thirds of the 120 seats in the state Legislature and, in the person of Jerry Brown, the governor’s office.

Even so, Republicans here sense a golden chance to change their short-term fortunes a few months before Election Day. Their optimism arises from a ballot measure to repeal a gas tax hike enacted by legislators last year to boost funding to repair California’s decaying roads and bridges. In last month’s primary election, Republicans stoked voter discontent over the 12-cent-a-gallon tax and recalled Josh Newman, a Democratic state senator from Orange County who supported the increase.

The recall of Senator Newman and recent polls showing strong opposition to the tax among conservatives and independent voters have roused Republicans in a manner last seen in 2003, during the recall of Gov. Gray Davis, a Democrat. Jason Roe, a Republican strategist based in San Diego, predicts that the repeal initiative will lure centrists and moderate Democrats to the right in November. Such a shift could provide a lift to Republican candidates in seven congressional districts that Democrats hope to flip as they seek to regain control of the House.

“The state’s voters are not as far left-of-center as people might imagine, and the gas tax is an issue that aligns with working-class values,” Mr. Roe says. He adds that persistent criticism of President Trump from California’s Democratic leaders has sharpened the ire of conservatives. “Their frustration will be clear on Nov. 6.”

As California confronts a $130 billion backlog in roadway and bridge repairs, the repeal measure, or Proposition 6, sets up a collision between progressive ideals and pocketbook politics in a state as known for its Democratic dominance as its high taxes.

The initiative arrives as the average price of gas in California hovers around $3.66 a gallon, the nation’s costliest after Hawaii, while the state’s gas tax of 55 cents a gallon ranks second-highest behind Pennsylvania’s. Supporters of Prop. 6 intend to amplify those numbers as they attempt to drown out the reasons Democratic lawmakers and Governor Brown pushed for the state’s first gas tax hike since 1994.

The state’s almost 400,000 miles of roadways rate among the country’s worst, with 37 percent considered in poor condition and another 42 percent graded as mediocre or fair. Traveling on deficient roads costs California motorists an estimated $53.6 billion a year in added vehicle operating expenses, traffic delays, and crashes, including an average of $2,450 per driver in the five largest urban areas. The tax hike and related fees would raise $5.4 billion a year over the next decade to upgrade roads and bridges and expand and enhance mass transit systems.

A coalition of local governments, labor unions, business leaders, and environmental groups has vowed to fight the repeal, aided by Brown, who will retire when his term ends in January. The unlikely alliance of special interests illuminates the unusual ideological clash wrought by Prop. 6.

“The repeal campaign could show the limits of California’s liberalism,” says Thad Kousser, a political science professor at the University of California, San Diego. “But that’s because it’s bad for progressives to defend a regressive tax. This is a tax that hits the poor harder than the rich. That tears right at the heart of the Democratic Party.”

Appealing to ‘Jekyll-and-Hyde voters’

California voters approved a ballot initiative in 2004 that levied a 1 percent tax surcharge on annual incomes above $1 million to generate funding for mental health services. Six years ago, they voted in favor of raising state income taxes on earnings of more than $250,000, and later supported extending the increase.

The measures passed with relative ease in a state that collects almost half its income taxes from the wealthiest 1 percent. The gas tax hike, by contrast, has drawn criticism as an excessive burden on working-class residents by further reducing their disposable income.

“The far left sees the tax as good because it affects behavior,” Roe says. “Their thinking is, ‘If it becomes more expensive to drive to work, you’ll find an alternative to driving.’ That doesn’t help people who live where there are no alternatives or who can’t afford more fuel-efficient cars.”

John Cox, the Republican candidate for governor, has turned Prop. 6 into a campaign refrain in his long-shot bid against Lt. Gov. Gavin Newsom (D), his opponent, who backs the gas tax. The repeal effort has raised more than $2 million and received donations from prominent Republicans, among them House Speaker Paul Ryan of Wisconsin and Kevin McCarthy, the House majority leader from Bakersfield, Calif.

Political analysts regard Prop. 6 as a kind of Trojan horse that holds the larger aspirations of Republican leaders in Washington as the party tries to a thwart a “blue wave” and preserve its majority in Congress. In the view of David McCuan, a political scientist at Sonoma State University, the measure could improve the odds of the party’s congressional candidates prevailing in tight races.

“Republicans are irrelevant in Sacramento, so they use the ballot box to get things in front of voters,” he says. “Ninety-five percent of a ballot measure’s success comes down to timing. With gas prices up, that puts the anti-tax folks in the driver’s seat.”

Dr. McCuan asserts that the potential impact of Prop. 6 depends on what he dubs the “Jekyll-and-Hyde voters” in the political middle whose party loyalty oscillates as they weigh conflicting desires. “These are the people who want good schools, good roads, and good jobs,” he says, “but they don’t want to pay for it.”

Mr. Cox, seizing on a populist crusade, has accused Democrats of “ignoring the needs of working Californians.” The rhetoric emphasizes high gas prices while skirting the less obvious costs of crumbling roads, explains Ethan Elkind, author of “Railtown,” a history of the Los Angeles metro rail system.

“The repeal may be short-sighted, but ‘gas tax’ just doesn’t sound good,” says Mr. Elkind, director of the climate program at the University of California, Berkeley School of Law. “If you drive over a pothole and break an axle, that’s going to set you back more than having to pay 12 cents more for a gallon of gas. But it’s harder to imagine that happening than to see what you’re paying every time you go to the pump.”

A progressive sales pitch

The ballot measure’s influence beyond Republican voters could hinge on the extent to which the election serves as a referendum on the president. In the state leading the anti-Trump resistance, and where Mrs. Clinton carried the seven congressional districts that national Democrats hope to wrest from Republicans, the president’s approval rating remains below 40 percent.

Former Democratic strategist Darry Sragow expects Trump’s looming presence to boost turnout statewide compared with the 2014 midterms, when 42 percent of registered voters went to the polls, a record low. (Turnout for last month’s primary election topped the 2014 primary by 12 percentage points.) Yet he ascribes support for Prop. 6 among independents and moderate Democrats to concern that the tax hurts the poor, and he doubts that voting for one conservative-backed measure would induce them to cast ballots for Republican candidates.

“Every once in a while there’s an initiative that becomes a lightning rod,” says Mr. Sragow, who publishes the California Target Book, a nonpartisan election and campaign guide that compiles statewide voter data. “But I don’t see support for the repeal as cracks in the progressive coalition. There clearly are going to be some progressives who feel that the gas tax falls more heavily on people with lower incomes and want to get rid of it.”

Groups working to defeat the ballot measure have amassed more than $11 million for the cause, and Brown has promised to raise as much as $25 million. He has enlisted local officials to tout the allocation of $9.2 billion in existing and future revenue from the gas tax for dozens of road and transit projects, ranging from carpool lanes in Sacramento to light-rail extensions in Los Angeles County.

Brown’s tactics reveal his awareness that the measure’s passage would imperil dozens of projects already under way and diminish his legacy. Sragow contends that Brown needs to deliver a progressive sales pitch, framing the tax as a shared responsibility that, over time, will improve roads while saving money for commuters and the state alike.

“If the tax is going to survive,” Sragow says, “the governor and his supporters have to start sending a very loud and credible message that it fills a critical need.”

During a public appearance days before the gas tax passed last year, Brown urged voters to pressure lawmakers to support the initiative. Referring to the state’s roads, he said, “Fix them now or you may never get them fixed.”

Longtime observers of Brown, who has also absorbed scorn from Republicans for his advocacy of a $77 billion high-speed rail project, believe he feels liberated as his political career nears its end. McCuan foresees the governor waiting until two months before the election to launch a counteroffensive against Prop. 6.

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“The advantage of moving second is you know your opponent’s strategy,” he says. The drawback might be that, with gas prices expected to reach as high as $4 a gallon this summer, voters will think only of their next visit to the pump when they head to the polls.

“Democrats have reason to be concerned,” McCuan says, “because the pocketbook trumps partisanship.”