Sen. Rob Portman (R) greets people before participating in a "Paddling For Patriots" event n Cleveland. (Jabin Botsford/The Washington Post)

When Hillary Clinton and her vice-presidential running mate, Tim Kaine, took a celebratory bus tour through crucial Ohio after the Democratic National Convention, they got some unexpected company: Republican Sen. Rob Portman.

Portman’s embattled reelection campaign had dispatched a squad of volunteers to Clinton-Kaine rallies in Columbus and Youngstown. There, they passed out literature touting his endorsements by several traditionally Democratic unions, signed up 400 new supporters and gathered more than 100 requests for yard signs, said Corry Bliss, Portman’s campaign manager.

The campaign also featured Portman’s outreach to Clinton supporters on its Facebook and Twitter accounts.

Portman is betting that a significant number of Ohioans in this turbulent election season might do something voters have not done in a long time: divide their preferences between the two parties as they work their way down the ballot. Breaking that pattern may be key to the survival of some endangered Republicans and possibly to the GOP’s hopes of maintaining control of the Senate. It’s a clear acknowledgment of the fear that Republican nominee Donald Trump is pushing away some voters — and of the threat he poses to the rest of his party.

Former Ohio governor Ted Strickland, the Democratic nominee for the Senate in Ohio, speaks in Columbus, Ohio. (Jay Laprete/AP)

Voters like to insist that they cast their ballots on the basis of the candidate, not the party. And the largest single bloc of voters is the 39 percent who identify as independent, according to a study of 2014 data by the Pew Research Center.

But their behavior in the voting booth speaks differently.

Split-ticket voting, once common, has in recent elections been rare in this polarized country. In 2012, for instance, only 6 percent of congressional districts — just 26 out of 435 — went for one party in the presidential race and another in picking a House member.

It was the lowest rate in 92 years — and a far cry from the zenith of split-ticket voting, which happened in Richard M. Nixon’s landslide of 1972, when 44 percent of the districts in the country voted one way for president and the other for the House.

Ohio is a good example of the trend. It has not split its preferences for the White House and the Senate since 1988, when it voted for Republican George H.W. Bush and to reelect then-Sen. Howard Metzenbaum (D).

There are some signs that Portman may be succeeding. The latest NBC News-Wall Street Journal-Marist poll, for instance, shows the senator holding a five-point lead over the Democratic nominee, former governor Ted Strickland, despite how Clinton has pulled ahead in Ohio by a similar margin.

A month ago, that same survey had the Senate and presidential races tied in the state.

Democratic presidential nominee Hillary Clinton leads her Republican opponent Donald Trump by 8 points in a new Washington Post/ABC News poll. The poll also found a majority of voters see both candidates as dishonest. (Bastien Inzaurralde/The Washington Post)

Republican incumbents are in tight races in six states that President Obama carried in 2012. In addition to the Ohio contest, those are Florida, Illinois, New Hampshire, Pennsylvania and Wisconsin.

Democrats need five seats to take back a Senate majority — or four, if they also hold onto the White House, giving Kaine a tie-breaking vote in the chamber.

That both parties have nominated relatively unpopular candidates for president is the main force that could disrupt what has become the typical straight-ticket dynamic.

Trump has higher negative ratings than any standard-bearer in history; were he not in the race, that dubious distinction would go to Clinton. Also scrambling the equation is how more and more leading Republicans are turning their backs on Trump.

GOP senators on the ballot this year are, by and large, performing better than Trump in the polls. They have their own organizations and bases of support.

“The unfavorable levels at the top of the ticket set up a condition that might enhance more ticket splitting than we have seen in recent elections,” said Christopher Borick, director of the Muhlenberg College Institute of Public Opinion in Pennsylvania.

In his own state, Borick noted, Clinton appears to have a double-digit lead over Trump in the latest surveys, but the battle between Sen. Patrick J. Toomey (R) and Democrat Katie McGinty is a dead heat.

“Pat Toomey seems to be holding on better to Republicans and winning more swing voters than Donald Trump,” Borick added. “If Trump becomes so unacceptable to a number of Republicans that they can’t vote for him, that might become a scenario where ticket-splitting perks up a bit.”

What they cannot afford, however, is for the bottom to drop out from under their presidential nominee.

In New Hampshire, on the other hand, Trump’s unpopularity appears to be dragging down the reelection prospects of Sen. Kelly Ayotte (R), despite her efforts to distance herself from him and his caustic comments about her.

The latest WBUR poll shows Trump running 15 points behind Clinton in the Granite State and Ayotte doing almost as poorly, with a 10-point deficit against Gov. Maggie Hassan (D).

Voters overall may be more unsettled — and therefore, up for grabs — than in the recent past.

An average of national polls taken in July, for instance, showed that 12 percent of the electorate had not made up its mind between Trump and Clinton — a higher share of undecideds at that point in the cycle since 1992, said Karl Rove, who was then-President George W. Bush’s chief political strategist.

“We’re going to see a larger group of voters in play than we have before,” said David Winston, a Republican pollster and longtime adviser to the congressional leadership. “We’ll have to reach these campaigns that are not used to ticket-splitting, and teach them how to do it.”

Some Democrats, however, are skeptical, especially given Trump’s stumbles since the convention, and the growing numbers of Republican leaders who are saying they will not vote for him.

“I don’t think [Trump] diminishes the numbers of swing voters, but his inability to speak beyond the base of his primary electorate has put him in a corner,” said Joel Benenson, Clinton’s pollster and chief strategist.

House Republicans are already appealing to voters to cast their ballots for Republicans in Congress as a brake on Clinton.

“If we fail to protect our majority in Congress, we could be handing President Hillary Clinton a blank check,” House Speaker Paul D. Ryan (R-Wis.) wrote in an urgent fundraising appeal earlier this month. The awkward implication was that Trump is unlikely to win.

That kind of calculation — in which ticket-splitting becomes a kind of check and balance — is known as “strategic voting.”

“It rarely happens — almost never happens — but this year is such an unusual situation that you could actually imagine it happening,” said Mark Mellman, a Democratic pollster who is working on gubernatorial and Senate races nationwide.

He added, however, that the dynamic is not likely to become clear until late in the campaign, possibly in the final weeks of October. And there remains the potential for voters to be so turned off by the presidential nominees that many decide to stay home.

Meanwhile, the effort to poach across party lines is working both ways.

In Ohio, for instance, Strickland campaign spokesman David Bergstein called Portman’s hopes for a split-ticket path to victory a “fantasy strategy.”

But Strickland will be reminding Trump voters about Portman’s record of supporting free trade.

“By the time this election is over, every voter in Ohio, across Appalachia and the Mahoning Valley, will know that Portman is the best senator China has ever had,” Bergstein said.

Portman, meanwhile, is boasting of his endorsement by unions that traditionally support Democrats, including the Ohio Teamsters.

“This year, our endorsements are all over the map,” said Fred Crow, political coordinator for the 2,800-member Teamsters Local 436 in northeast Ohio.

Among the rank and file, “I think there is going to be a lot of ticket-splitting,” Crow added.