President Obama’s February 2009 stimulus bill, the $787 billion American Recovery and Reinvestment Act, was a political disaster.

It helped fuel the Republican revival of 2010 and now stars in Mitt Romney’s ads. The president even stopped uttering the word “stimulus.” But the bill was one of the most important and least understood pieces of legislation in modern history.

It was the purest distillation of what Obama meant by change, transforming our approaches to energy, education, health care, transportation and the economy, promoting long-term reinvestment as well as short-term recovery. Just about everything Americans think they know about it is wrong. Here are a few examples.

1. The stimulus didn’t create jobs.

A year after Obama signed the bill, the percentage of the public that believed it had created jobs was lower than the percentage that believed Elvis was alive. But at its peak, the Recovery Act directly employed more than 700,000 Americans on construction projects, research grants and other contracts.

That number doesn’t include the jobs saved or created through its unemployment benefits, food stamps and other aid to struggling families likely to spend it; its fiscal relief for cash-strapped state governments; or its tax cuts for more than 95 percent of workers. Top economic forecasters estimate that the stimulus produced about 2.5 million jobs and added between 2.1 percent and 3.8 percent to our gross domestic product.

The stimulus didn’t keep unemployment below 8 percent, as the Obama team predicted in an ill-advised report designed to help pass the bill. Unemployment soared past 8 percent before the stimulus even kicked into gear. It later became clear that the economy was free-falling much faster than experts realized at the time; the initial GDP estimate for the fourth quarter of 2008 was a recession-level negative 4 percent, later revised to a depression-level negative 8.9 percent.

But, as I detail in my new book on the stimulus, “The New New Deal,” the bill helped stop that free fall. Job losses peaked the month before it passed. The jobs numbers that spring, while grim, marked the biggest quarterly improvement in almost 30 years. The Recovery Act launched a weak recovery, but even a weak recovery beats a depression.

2. The stimulus was full of waste, pork and fraud.

Most of the Recovery Act consisted of straightforward aid to states and to the vulnerable, infrastructure spending, and tax cuts. Critics may call it “porkulus,” but the stimulus was also the first modern spending bill with no official legislative earmarks, the usual definition of “pork.” And after experts warned that 5 to 7 percent of the money could be lost to fraud, investigators documented only $7.2 million in losses through 2011, about 0.001 percent.

Of course, waste is in the eye of the beholder. But it’s telling that most Republican examples of stimulus boondoggles were either removed from the bill (sod on the National Mall, smoking cessation), never in the package (mob museums, levitating trains to Disneyland) or wild distortions (The new Department of Homeland Security headquarters is not “government furniture”).

Yes, there was Solyndra, but its $535 million default represented only about 1 percent of the Recovery Act’s clean-energy loan portfolio, and independent reviewers have found that the overall portfolio is in fine shape. And Republican investigators have found no evidence that cronyism drove the Solyndra loan.

3. The stimulus should have been much bigger.

It’s true that a bigger stimulus would have provided a bigger economic jolt and accelerated the sluggish recovery. More public works projects would have produced more jobs; more state aid would have prevented more of the public-sector cutbacks that have slowed the recovery; more tax cuts would have directed more cash into your wallet. But there was no way Obama could have gotten another dime out of Congress.

The bill needed 60 votes in the Senate to overcome a Republican filibuster, and the three GOP moderates who supported it insisted that it couldn’t exceed $800 billion. So did at least half a dozen centrist Democrats, including Ben Nelson (Neb.), Mark Begich (Alaska) and Mary Landrieu (La.). Everyone involved in the negotiations — including liberals who favored a larger stimulus — agrees that Obama got as much as he could get.

Remember: Just five months earlier, a $56 billion bill died in the Senate. After Obama’s election, 387 liberal economists urged Congress to approve a $300 billion to $400 billion package. It’s only in retrospect that the final amount — larger than the entire New Deal in constant dollars — seems modest.

4. Unlike the New Deal, the stimulus will leave no legacy.

Nostalgic liberals often complain that the stimulus lacks iconic Hoover Dams and Skyline Drives. In fact, it’s creating its own icons: zero-energy border stations, state-of-the-art battery factories, some of the world’s largest wind farms and half a dozen of the largest solar farms. But its main legacy, like the New Deal’s, will be change.

The stimulus was the biggest and most transformative energy bill in history, pouring an astonishing $90 billion into record expansions of every imaginable form of clean energy, from renewables to electric vehicles. It included $27 billion to computerize health care. Its Race to the Top was a landmark in education reform. Its high-speed rail program was the most ambitious transportation initiative since the interstates. It extended high-speed Internet to underserved communities, a modern twist on the New Deal’s rural electrification, and modernized the New Deal-era unemployment insurance system. And much more.

America didn’t need a new Hoover Dam, although the bill did finance the largest dam-removal project ever. The stimulus will leave a different legacy: a down payment on a greener, more competitive economy with a healthier, better-educated, better-connected workforce.

5. The stimulus showed that Obama can’t legislate.

Obama is often criticized for punting the stimulus to Capitol Hill. But while Congress wrote the bill, the president dictated its principles and most of its specific content. Its major initiatives on tax cuts, energy, education, health care and the economy came straight out of his campaign agenda. The final bill emerged from a list of spending items drafted by the White House.

Obama didn’t get everything he wanted. For example, Republican Sen. Susan Collins (Maine) killed his plan for a school construction binge. But the president needed her vote. The deals that got the Recovery Act done served notice that after campaigning as a change-the-system outsider, Obama would govern as a work-the-system insider. He was pragmatic enough to recognize that a bill that can’t pass Congress can’t make change.

Michael Grunwald, a former Washington Post reporter, is a senior national correspondent at Time magazine and the author of “The New New Deal: The Hidden Story of Change in the Obama Era.”