In promoting the health care law, President Barack Obama is repeating his persistent and unsubstantiated assurance that Americans who like their health insurance can simply keep it. Republican rival Mitt Romney says quite the opposite, but his doomsday scenario is a stretch.

After the Supreme Court upheld the law last week, Obama stepped forward to tell Americans what good will come from it. Romney was quick to lay out the harm. But some of the evidence they gave to the court of public opinion was suspect.

A look at their claims and how they compare with the facts:

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OBAMA: “If you’re one of the more than 250 million Americans who already have health insurance, you will keep your health insurance. This law will only make it more secure and more affordable.”

ROMNEY: “Obamacare also means that for up to 20 million Americans, they will lose the insurance they currently have, the insurance that they like and they want to keep.”

THE FACTS: Nothing in the law ensures that people happy with their policies now can keep them. Employers will continue to have the right to modify coverage or even drop it, and some are expected to do so as more insurance alternatives become available to the population under the law. Nor is there any guarantee that coverage will become cheaper, despite the subsidies many people will get.

Americans may well end up feeling more secure about their ability to obtain and keep coverage once insurance companies can no longer deny, terminate or charge more for coverage for those in poor health. But particular health insurance plans will have no guarantee of ironclad security. Much can change, including the cost.

The non-partisan Congressional Budget Office has estimated that the number of workers getting employer-based coverage could drop by several million, as some workers choose new plans in the marketplace or as employers drop coverage altogether. Companies with more than 50 workers would have to pay a fine for terminating insurance, but in some cases that would be cost-effective for them.

Obama’s soothing words for those who are content with their current coverage have been heard before, rendered with different degrees of accuracy. He’s said nothing in the law requires people to change their plans, true enough. But the law does not guarantee the status quo for anyone, either.

So where does Romney come up with 20 million at risk of losing their current plans?

He does so by going with the worst-case scenario in the budget office’s analysis. Researchers thought it most likely that employer coverage would decline by 3 to 5 million, but the range of possibilities was broad: It could go up by as much as 3 million or down by as much as 20 million.

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ROMNEY: After saying the new law cuts Medicare by $500 billion and raises taxes by a like amount, adds: “And even with those cuts and tax increases, Obamacare adds trillions to our deficits and to our national debt, and pushes those obligations onto coming generations.”

THE FACTS: In its most recent complete estimate, in March 2011, the Congressional Budget Office said the new health care law would actually reduce the federal budget deficit by $210 billion over the next 10 years. In the following decade, the law would continue to reduce deficits by about one-half of one percent of the nation’s gross domestic product, the office said.

The congressional budget scorekeepers acknowledged their projections are “quite uncertain” because of the complexity of the issue and the assumptions involved, which include the assumption that all aspects of the law are implemented as written. But the CBO assessment offers no backup for Romney’s claim that the law “adds trillions to our deficits.”

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OBAMA: “And by this August, nearly 13 million of you will receive a rebate from your insurance company because it spent too much on things like administrative costs and CEO bonuses and not enough on your health care.”

THE FACTS: Rebates are coming, but not nearly that many Americans are likely to get those checks and for many of those who do, the amount will be decidedly modest.

The government acknowledges it does not know how many households will see rebates in August from a provision of the law that makes insurance companies give back excess money spent on overhead instead of health care delivery. Altogether, the rebates that go out will benefit nearly 13 million people. But most of the benefit will be indirect, going to employers because they cover most of the cost of insurance provided in the workplace.

Employers can plow all the rebate money, including the workers’ share, back into the company’s health plan, or pass along part of it.

The government says some 4 million people who are due rebates live in households that purchased coverage directly from an insurance company, not through an employer, and experts say those households are the most likely to get a rebate check directly.

The government says the rebates have an average value of $151 per household. But employers, who typically pay 70 to 80 percent of premiums, are likely to get most of that.

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ROMNEY: “Obamacare raises taxes on the American people by approximately $500 billion.”

THE FACTS: The tax increases fall heavily on upper-income people, health insurance companies, drug makers and medical device manufacturers.

People who fail to obtain health insurance as required by the law will face a tax penalty, although that’s expected to hit relatively few because the vast majority of Americans have insurance and many who don’t will end up getting it. Also, a 10 percent tax has been imposed on tanning bed use as part of the health care law. There are no other across-the-board tax increases in the law, although some tax benefits such as flexible savings accounts are scaled back. Of course, higher taxes on businesses can be passed on to the consumer in the form of higher prices.

Individuals making over $200,000 and couples making over $250,000 will pay 0.9 percent more in Medicare payroll tax and a 3.8 percent tax on investments. As well, a tax starts in 2018 on high-value insurance plans.

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OBAMA: “Because of the Affordable Care Act, young adults under the age of 26 are able to stay on their parents’ health care plans, a provision that’s already helped 6 million young Americans.”

THE FACTS: Obama is overstating this benefit of his health law, and his own administration knows better. The Department of Health and Human Services, in a June 19 news release, said 3.1 million young adults would be uninsured were it not for the new law. Obama’s number comes from a June 8 survey by the Commonwealth Fund, a health policy foundation. It said 6.6 million young adults joined or stayed on their parents’ health plans who wouldn’t have been able to absent the law. But that number includes some who switched to their parents’ plans from other coverage, Commonwealth Fund officials told the Los Angeles Times.

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ROMNEY: “Obamacare is a job-killer.”

THE FACTS: The CBO estimated in 2010 that the law would reduce the amount of labor used in the economy by roughly half a percent.

But that’s mostly because the law will give many people the opportunity to retire, stay at home with family or switch to part-time work, since they will be able to get health insurance more easily outside of their jobs. That voluntary retreat from the workforce, made possible by the law’s benefits, is not the same as employers slashing jobs because of the law’s costs, as Romney implies.

The law’s penalties on employers who don’t provide health insurance might cause some companies to hire fewer low-wage workers or to hire more part-timers instead of full-time employees, the budget office said. But the main consequence would still be from more people choosing not to work.

Apart from the budget office and other disinterested parties that study the law, each side in the debate uses research sponsored by interest groups, often slanted, to buttress its case. Romney cites a Chamber of Commerce online survey in which nearly three-quarters of respondents said the law would dampen their hiring.

The chamber is a strong opponent of the law, having run ads against it. Its poll was conducted unscientifically and is therefore not a valid measure of business opinion.