If President Obama wins a second term, he will finally endorse same-sex marriage. Gay rights groups are almost certain. He will also make a new, historic effort to fight climate change — environmentalists are pretty sure.

And Obama will finally do just what the Congressional Black Caucus wants. According to some members of the Congressional Black Caucus.

Conservative groups are equally confident that Obama, freed from the fear of losing his reelection bid, would deliver on far-reaching left-wing dreams. GOP candidate Mitt Romney forecasts a runaway spending spree. Newt Gingrich envisions a “war” on the Catholic Church. The National Rifle Association predicts a crackdown on gun owners.

The funny thing about all this is: Obama himself hasn’t said he’ll do any of it.

In his speeches — over the first few months of his re­election campaign — the president has only sketched out a vague agenda for his next term. He wants to fix the immigration system. Put his health-care law into practice. Rebuild infrastructure. Revive manufacturing.

“And,” he told an audience in San Francisco, in what might be called a flourish of the obvious, “we’re going to have to figure out how to pay for all this stuff.”

This disconnect highlights one of the most unusual factors in an unusual campaign: Even after three years in office, Obama remains a political Rorschach test. His friends still project their brightest hopes on him. His enemies still project their deepest nightmares.

Both are still convinced they haven’t seen the real Obama yet — or the real Obama agenda.

And so, paradoxically, they believe the most important ideas of Obama’s reelection campaign are the ones he’s not talking about.

“All that first-term lip service to gun owners is part of a massive Obama conspiracy to deceive voters,” NRA chief executive Wayne LaPierre said during the Conservative Political Action Conference. “And hide his true intentions to destroy the Second Amendment during his second term!” His evidence was Obama’s past views on gun control, and his appointment of two Supreme Court justices the NRA considers hostile.

The speculation about a yet-unseen Obama agenda reflects an unusual commonality between left and right in a polarized election year. It also points to a strategy each side will use this year — trying to marshal excitement among core supporters by tapping into deep pools of worry or hope about Obama.

Obama’s own campaign says there’s nothing hidden here. They say the president has already laid out much of his vision for a second term.

“There shouldn’t be any mystery around the president’s agenda, whether it’s today or three years from now,” said Obama campaign spokesman Ben La­Bolt. He said Obama “has outlined a vision for an economy that’s built to last, one where hard work pays, responsibility is rewarded and everyone plays by the same set of rules.”

For details, aides point to Obama’s State of the Union address this year and his budget proposal. These called for transforming the tax code to add taxes on the rich — the “Buffett Rule.” Obama also proposed making a college education more attainable and boosting American manufacturing.

But, at times, the details of Obama’s vision can still appear fuzzy — even to him.

“How do we make sure that we’ve got a tax system that reflects everybody doing their fair share?” Obama asked at a Jan. 31 fundraiser in Washington, describing his vision in terms of questions, not answers. “How do we make sure that we’ve got a set of regulations in place that aren’t designed to squelch entrepreneurial activity and the free market but are designed to make sure that our consumers are protected and that our air is clean?”

“Those are going to be huge fights,” Obama added.

The GOP candidates running to replace Obama have released dozens of specific proposals: for cutting taxes, for reducing Medicare spending, even for a moon colony. Obama’s allies, instead, look for hints and nods.

For instance: Obama officially opposes same-sex marriage. But, at a gay-pride event in the East Room last June, Obama noted that those in the room were fighting for the rights of parents and students and partners.

He paused, seemingly for emphasis. “ . . . and spouses,” Obama said. He paused again, for the cheers and whoops to subside.

“I have delivered on what I promised,” the president said. “Now, that doesn’t mean our work is done.”

To gay-rights advocates, the meaning is obvious. “It’s the next logical step” for Obama to announce his support of same-sex marriage, said Rebecca Isaacs, executive director of the Equality Federation.

Obama has also said that he will push for immigration reform. “My presidency is not over. I’ve got another five years coming up. We’re going to get this done,” he told Univision radio host Eddie “Piolin” Sotelo last month.

But, in his campaign talks, the president has not given many details about how he would do it. In the past, he has called for requiring illegal immigrants to get right with the law, submitting to background checks, and paying fines and back taxes, before they get on a path to legal status.

And inside the Congressional Black Caucus, Rep. Frederica Wilson (D-Fla.) says she’s also picked up on an agenda that Obama won’t talk about openly.

Last year, when some of Wilson’s colleagues in the black caucus began accusing Obama of not doing enough to help his fellow African Americans, Wilson stood at a town hall meeting to argue that Obama was committed to their issues — but that he could not tackle them so overtly until he got safely past Election Day.

“He has a lot to lose now” by doing so, said an Obama supporter who envisions a second term focused in part on help for inner-city schools and targeted dollars to stem an AIDS epidemic in black communities. “You have to anger a lot of people to do some of these things that I’m suggesting.”

On the campaign trail, Republican candidates have also described their own scenarios for another four years of Obama.

On their side, of course, the possibilities range from dour to near-apocalyptic.

“We hit a Greece-like wall,” Romney told TV host Bill O’Reilly last year. As evidence, a Romney aide cited Obama’s inability to tackle the growth in big-spending programs such as Medicare and Social Security.

“I think this country would be a fundamentally different country at that point,” former senator Rick Santorum (Pa.) said last month on an online show hosted by Glenn Beck. He envisioned greater government interference in everyday life — a transition to “European socialism, at a minimum.” “I think we will have lost the very essence of what America is about.”

If Obama does win, however, both the dreams and the nightmares may turn out to be overblown.

To enact the changes liberals want on taxes, immigration or gun laws, he would need a strongly liberal Congress. And that’s not likely.

“I haven’t seen any evidence that he’s going to do anything in his second term that isn’t defensive,” said Andrew Roth of the fiscally conservative Club for Growth. “He’s got to defend Obama­care, and try and implement it. Against, you know, a lot of opposition. He’s got to defend Dodd-Frank [financial regulation], against a ton of opposition.”

The stifling gridlock of Obama’s last year, he said, is not likely to vanish in 2013.

“Whoever is the president,” Roth said, “will have a very tight leash.”