WASHINGTON — This is presidential? They bicker, interrupt, talk over the moderator.

To some, the Obama-Romney rematch was squirm-inducing. But shedding some dignity probably won’t cost the candidates much. Since both President Barack Obama and challenger Mitt Romney came out swinging, neither was likely to claim a decisive advantage among viewers who thought the debate smacked of the wrong type of reality TV.

And many backers who were already lined up on the two sides of the super-heated race were looking for a scrappy face-off.

“In the world of ‘The Real Housewives,’ everybody needs to turn over a table from time to time,” said Evan Cornog, dean of the communications school at Hofstra University, where Tuesday night’s debate took place.

“How good that is for the republic,” he said, “I don’t know.”

The presidency isn’t a person, it’s an institution. And Americans traditionally expect presidents seeking re-election to maintain a certain level of decorum.

Challengers get more leeway but still are expected to pay deference to the office of chief executive, if not to the man. Maybe that tradition is doomed in a conflict-addicted popular culture where even television cooking shows are “throwdowns.”

Can the notion of the dignity of office survive the era of flash analysis, when a phrase like “binders full of women” launches a thousand Internet jokes — while the debate still is in progress — and campaigns spin the matchup into attack ads within hours?

The tone of Tuesday’s faceoff was embraced by Democrats who were dismayed by Obama’s dreary performance in the first debate. They had urged him to adopt a more brass-knuckles style.

When Obama stepped up to meet Romney’s hard-charging persona, the result was a presidential campaign matchup that stands out as one of the most rancorous on live TV, especially for an event in which the candidates were onstage with everyday folks, fielding their earnest questions.

Whether that was good or bad, it was one of the most exciting to watch.

Romney turned to the president and posed his own accusatory questions, demanding answers. When Romney made a point, Obama would shoot back, “Governor, that’s not true.”

Six times he declared Romney’s words “not true.”