It’s four years, two months and millions of rancorous words too late, but could President Barack Obama’s outreach to congressional Republicans be the start of something big?

By big, of course, I mean serious efforts to reach a grand bargain on the national debt — followed, maybe, by further bargains on immigration and steps needed to get the economy working for people besides big bankers and investors in the stock market.

Big moves would give Obama a chance to achieve a legacy of positive achievement — maybe, even, of his original stated purpose of closing the partisan chasm dividing Americans.

If this is what he had in mind by going to dinner with 12 GOP senators, having lunch with House Budget Chairman Paul D. Ryan and scheduling visits to Capitol Hill next week, it would be a total reversal of his past pattern of aloofness richly mixed with hostility.

For months, he’s been vilifying Republicans as uncaring about disabled children and the aged, interested only in protecting tax breaks for oil companies and private jet owners and willing to force schoolteachers, Border Patrol agents and first responders out of their jobs.

He waited until the Friday before the sequester took effect to meet with Speaker John A. Boehner and Senate Minority Leader Mitch McConnell, when it was a foregone conclusion that $85 billion in what he later called “dumb, arbitrary cuts” would take effect.

At a news conference just five days before the dinner, he repeated estimates that the sequester might cost 750,000 jobs and said: “None of this is necessary. It’s happening because of a choice that Republicans in Congress have made. They’ve allowed the cuts to happen because they refuse to budge on closing a single wasteful loophole … for the well-off and well-connected.”

So, the first test of whether Obama has really changed strategies (not tactics) will be: Does he stop the attacks? Does his super PAC, Organizing for America, promote his agenda positively or spend its millions of dollars demonizing opponents?

The second test is: Does the outreach continue or — at the first sign of a roadblock — does he say, “See, I tried to work with those people. It’s useless.”

If that happens, it’ll be a sign that Obama’s real strategy — as Republicans suspect — is to further divide them in hopes of capturing the House in 2014 and pushing his liberal agenda through in the last two years of his presidency.

That’s another way to achieve a legacy, but it’s a big gamble, given the GOP advantage in gerrymandered House districts.

The third and biggest test will be on substance: Does he produce proposals to revise both entitlements and the tax system?

Does he, as one of his dinner companions said, “stand at the podium and tell the country that Medicare recipients pay in only a third of what they take out and that structural reforms are necessary, not just tinkering?”

He gave the GOP senators the impression that he understands the need. “The question is, will he lead?” one of them asked.

Of course, it takes two sides to have a grand bargain. “We know he wants more revenue,” one senator told me. “There are a lot of us ready to give it to him through a tax reform that also lowers rates, if we get entitlement reform, too. We said so.”

“From a dinner to a deal is a long and winding path,” this senator said. “The next five months give us the best chance we’ll have. But he’s got to lead.”

It’s not at all clear why Obama changed strategies. Maybe his polls showed that sequester scare tactics were not working. But I’d  hope it was recognition that there always has been a “caucus of common sense” and that he ought to mobilize it.