John Kasich and Donald Trump. David A. Grogan | CNBC

President Donald Trump and Ohio Gov. John Kasich are both Republicans, but they are often at odds with each other. Yet, at least for the next few days, they share a goal: helping GOP candidate Troy Balderson avoid losing a longtime Republican House seat on the outskirts of Columbus, Ohio. The Ohio governor endorsed Balderson for Tuesday's highly contested Ohio 12th District special election, calling him a "partner in turning around Ohio as we passed tax cuts and balanced the budget." Trump swooped in Saturday night to try to give the state senator Balderson a last-second boost, backing him during a raucous rally in Delaware, Ohio. On Monday, Balderson told reporters that the endorsements show his campaign has "unified the Republican Party" as he looks to beat Democrat Danny O'Connor in a close race for a typically safe GOP seat that Trump carried by 11 percentage points in 2016. If O'Connor can win in the red-leaning area that is more wealthy and highly educated than the typical congressional district, observers see a bad sign for the GOP as it tries to defend suburban swing seats and stop Democrats from taking a House majority in November. The president and governor's rare moment of consensus may help the GOP avoid another embarrassing special election loss similar to Democratic Rep. Conor Lamb's triumph in a red piece of southwestern Pennsylvania earlier this year. But the fact that Kasich and Trump's agreement is such a rare event also underscores how much the Republican base, even in Ohio, has evolved in Trump's image. "I think [Kasich is] not a true Republican, not a true conservative. ... I didn't like the way he didn't support President Trump," said Hassan Dakhteh, the owner of Stogies Cigar Lounge in Powell, a city north of Columbus. The Republican said he supports Balderson for Congress, mainly because he thinks the candidate will support Trump's agenda.

O'Connor aims to take advantage of Kasich, Trump differences

The vast majority of Republicans have coalesced around Trump and nearly all of his policies. But frustration with Trump among some Republicans, embodied by Kasich, has given an opening to O'Connor and other Democrats trying to win GOP-leaning districts this year. O'Connor, the 31-year-old Franklin County recorder, is trying to win over middle-of-the-road voters who feel put off by the GOP's turn toward Trump. Kasich's endorsement of Balderson may make those centrist voters more likely to vote Republican. But O'Connor on Tuesday downplayed Kasich's support for his opponent: "Sun rises in the East, sets in the West, Republican endorses Republican," he told CNBC. Still, the Democrat made a point to highlight the policies on which he agrees with the governor. Those include expansion of the federal and state Medicaid insurance program for low-income Americans and "red flag" laws that bar access to guns for people with a history of domestic violence or mental illness. "There are things that we can all agree on that move the country forward, and I'm glad that Kasich and I stand together on some of those," O'Connor said. O'Connor aims to win a district that Republicans have represented since Kasich himself first won the seat in 1982 at age 30. Democrats got an opening when Kasich's successor, GOP Rep. Pat Tiberi, resigned effective earlier this year to lead the Ohio Business Roundtable. O'Connor and Balderson will face off Tuesday for the right to fill the remainder of Tiberi's term into January, then run again in November for the next Congress.

The Kasich-Trump divide

Kasich has tried to cultivate a brand as a centrist fiscal conservative. That has set him at odds with Trump — and many of the Republican Trump voters who live in the area. Some of them consider Kasich too moderate and not supportive enough of the president.

Kasich broadly backs tax reform and free trade. He supports the federal and state Medicaid insurance for low-income Americans and moderate gun-control measures. The Ohio governor ran for president in 2016 and carried only his home state in the GOP primaries as Trump won the party's nomination. The president chided Kasich frequently about his primary record during the race, at one point calling him "1 for 42 John Kasich" in reference to the proportion of primaries he won. Kasich's allies have talked to donors about their interest in backing the governor should he challenge Trump in 2020, although the Ohio Republican has not made a decision on whether to run, CNBC reported earlier this year. Kasich has shredded many of Trump's policies. He has recently focused most of his ire on Trump's tariffs imposed on major trading partners China, Mexico, Canada and the European Union, calling the president's trade actions "terrible." Trump has dubbed his once — and potentially future — campaign rival a "failure" and a "dud" who is too soft on illegal immigration.

Many Republicans, even in Ohio, have echoed the president's criticism of the governor. No GOP voters who talked to CNBC in the 12th District last week said they would prefer Kasich over Trump in a hypothetical presidential primary. "Not unless he repents of his evil ways," Jeff Lutz said with a smile about whether he would vote for Kasich again. The Westerville, Ohio, man who works on personal computers wore a "Make America Great Again" hat and "Trump 2020" shirt Monday to a Newark, Ohio, rally where Vice President Mike Pence endorsed Balderson. Still, other voters who plan to support Balderson on Tuesday spoke positively about certain Kasich policies. A few GOP voters had particular praise for his support of Medicaid expansion.

Tight race down the stretch