“I think everybody understands how the insurance system works,” White House spokesman Jay Carney declared on Tuesday, without apparent irony.

A man holds a sign at the Tea Party Patriots 'Exempt America from Obamacare' rally on the west lawn of the Capitol on Sept. 10. (Jonathan Ernst/Reuters) A man holds a sign at the Tea Party Patriots ‘Exempt America from Obamacare’ rally on the west lawn of the Capitol on Sept. 10. (Jonathan Ernst/Reuters)

Actually, the public is really confused about the insurance system and what President Obama is trying to do to it. That’s partially — though far from entirely — the president’s fault. Nevertheless, next year’s may well be the third election in a row in which Americans will vote on Obamacare without really getting Obamacare.

About half to two thirds of people in recent polls admit to at least partially misunderstanding the reform. Other surveys show that support for reform changes when pollsters call it “the Affordable Care Act” than when they use the term “Obamacare.” The Post’s latest poll on the health-care law didn’t probe either of those matters, but widespread confusion about the law visible in the answers to the questions our pollsters did pose. Number after number in the poll should be flipped on its head.

The poll, for example, found that 65 percent of Americans oppose the law’s mandate that all individuals obtain insurance, whereas 58 percent favor its mandate that all medium and large employers offer coverage to their workers. These figures suggest that lots of people don’t object to government health-insurance mandates, but they think it should be up to employers to extend coverage to more people.

In reality, nearly all major employers already offer coverage, and the few who don’t would suppress wages and hiring if they have to provide insurance. It wouldn’t be free for those getting insurance. Even that, though, wouldn’t help the self-employed, the under-employed, the unemployed, or those who work for very small businesses. The individual mandate, not the employer provision, is the key to covering them, insisting that they pay what they can into a more rational system and enabling them to get quality, affordable health insurance in return.

The most striking number in the Post poll, though, is that 56 percent of Americans believe that the cancellation letters that insurance companies have been sending to some of their customers over the last several weeks aren’t “a normal startup problem with the new system,” but are rather “a sign of mismanagement of the new health-care law.” In fact, the law and the regulations that followed it have been clear for some time; many people would not be able to keep health-care plans that did not meet certain “essential benefits.” They would have to move onto better plans, many with government help. Just like the individual mandate, this transition is an important part of organizing the new health-care market for the people the employer-based system leaves out.

Obama bears a lot of blame for the confusion on the recent cancellations. He set expectations when he promised that Americans could keep their health-care plans. His promise didn’t bear out, which looks like incompetence, dishonesty or both. High complexity was also the byproduct of Democrats’ choice to keep most of the existing system intact and build around it.

But health-care reform was always going to be complicated — and easy for uncomplicated ideologues to denounce. Republican demagoguery — from the days of “death panels” to the current frenzy over cancellation letters — has done much more to confound public understanding of the Affordable Care Act, not to mention the law’s implementation.

Next year, voters should judge the law based on how many more people ultimately get affordable health-care coverage, and at what cost to taxpayers. Right now, many appear to be judging the law based on misunderstandings and warped representations.

Follow Stephen Stromberg on Twitter: @strombergsteve