Four years ago, Iowa launched President Obama toward the White House, and it has been a special place for him ever since. But he returned here Tuesday as an embattled incumbent trying to rally supporters to turn back a stiff challenge from Mitt Romney.

Republicans are surprisingly bullish about their candidate’s chances in a state where Obama has deep roots and where the unemployment rate is well below the national average. Democrats, including the president’s advisers, concede that this year’s campaign will be nothing like the relatively easy one Obama had four years ago.

Iowa fell hard for Obama in 2008. It was, as he said Tuesday, a state “that gave me a chance when no one else would.” That began with his victory in the Democratic caucuses, which helped him capture the nomination. He won the state by 10 percentage points against Sen. John McCain (R-Ariz.) in the general election.

Today, Iowa appears to have reverted to the kind of closely divided politics of the previous decade, when Vice President Al Gore won the state by less than a percentage point in 2000 and President George W. Bush followed with a similarly razor-thin victory in 2004.

Other battlegrounds are quite competitive, but Iowa stands apart. It is hotly contested despite the bond Obama established with many voters here four years ago. And unlike in some other battleground states, such as Virginia or Colorado, the elements of the Obama coalition that can provide some protection against the currents flowing this year — a high proportion of African American and Latino voters and a much younger and more mobile population — do not exist in Iowa.

Still, that Romney is as strong a challenger as he is may surprise some people. In contrast with Obama, there has been no love affair between Romney and the Hawkeye State. Iowa was where he suffered a bitter defeat in the 2008 Republican caucuses that crippled his candidacy, and the state’s GOP base is so conservative that the former Massachusetts governor spent much of last year weighing whether to make a serious run in the 2012 caucuses.

What has turned this state into one of the nation’s prime battlegrounds is a combination of dis­affection with the president, a weak national economy and the reality that Romney appears to be better suited to run in a general-election campaign than in caucuses dominated by activists who doubted his conservative convictions.

In his visit here Tuesday, Obama continued the rhetorical war with Romney and the Republicans that he started Monday over middle-class tax cuts. The president wants Congress to extend for all but the wealthiest Americans the George W. Bush-era tax cuts that are set to expire at the end of the year. Republicans want those cuts extended for all taxpayers.

To make his case, Obama visited a family in Cedar Rapids whose taxes could increase by $2,000 if the tax cuts expire, according to the president’s campaign, before speaking to a larger audience at Kirkwood Community College.

“My opponent and his allies in Congress, they sincerely believe that prosperity comes from the top down,” Obama told the overflow crowd. “They believe that if we spend trillions of dollars more in tax cuts, mostly for the wealthy, that somehow it will create more jobs.” He added, “And I think they’re wrong.”

In Colorado, Romney fired back during a town hall meeting in Grand Junction. Obama’s latest tax plan, he said, “added insult to injury with another kick in the gut.”

He said the president’s plan would keep taxes at the same level for many Americans while raising taxes on what he called “job creators and small businesses.” Romney cast the plan as “the sort of thing only an extreme liberal could come up with.”

Obama advisers have been uncertain about Iowa for many months. Part of the reason is the general disaffection for a president governing during a weak economy. In addition, after several tough elections, Republicans staged a comeback in the 2010 midterm elections and hope to extend that in November.

Republicans also point to a turnaround in voter registration in the state. Four years ago, Democrats held the clear advantage, but the numbers have shifted by more than 100,000. Republicans now lead Democrats by about 20,000.

The Obama campaign has built another extensive field operation in Iowa, with 18 offices across the state, far more than Romney has. As for the registration deficit, Obama campaign officials note that Iowa has same-day registration and that they plan to try to make up some of that gap closer to the election.

Another factor that has cost Obama is the fact that Republican presidential candidates spent many months and millions of dollars attacking him during their primary campaign in the fall and winter. “They dominated the conversation for a year,” Obama campaign manager Jim Messina said. “Once people focus on the issues, I think [Romney’s] going to have real trouble in Iowa.”

Iowa Democratic strategist Jeff Link said of Obama’s team: “They are aware and engaged. This has not caught anyone by surprise. They know the battle is joined. They know it’s tight. I think they wish they had a little more cushion, but they’re prepared.”

Link said Romney is “a better candidate in the general election than in the caucuses” because he is considered closer to the middle than many of the activists who participated in winter gatherings.

Iowa Gov. Terry Branstad (R) said Romney’s problems with the party’s conservative base have been diminished now that the opponent is the president. He said Obama’s support for same-sex marriage has helped galvanize evangelical conservatives behind Romney.

“He’s in trouble,” Branstad said of the president. “He knows he’s in trouble. He thinks the only way to do it is to smear Romney. But it isn’t going to work.”

Iowa’s economy is strong compared with the rest of the country, but strategists on both sides said national economic problems have a significant influence on voters’ attitudes. Still, Republicans said Romney must have a compelling case that he’s better prepared than Obama to turn things around.

They added that the GOP challenger should focus on debt and deficits, subjects of great importance to frugal Iowans. “Debt’s driving the issue here,” Branstad said. “Iowans hate debt.”

J. Ann Selzer, the leading independent pollster in the state, said Obama’s special relationship with Iowa extends beyond the Democratic activists who propelled him to victory in the caucuses, but how far that will go is unknown. She said the race is competitive because enough voters appear open.

“You can move people here, and you can move enough of them to change the election,” she said. “There is sort of an openness to another way out.”

Obama’s advisers believe the contrast in messages between the president and Romney will bring Iowa voters back to Obama’s side. But the president took every opportunity Tuesday to tell his supporters that the months ahead will be challenging.

“I’m betting that you are going to be as fired up as you were in 2008,” he said, “because you understand the stakes for America.”

Balz reported from Washington. Philip Rucker in Colorado contributed to this report.