Washington ? United on almost nothing, Barack Obama and John Boehner are the two faces of America’s divided government, the humbled president and the triumphant House leader. Both claim to speak for the people, yet they have had little to say to each other.

This is the relationship that will drive everything.

On first appearance, both men put on a public display Wednesday intended to emphasize what voters want: cooperation to create jobs. A reflective Obama acknowledged the drubbing his party took in Tuesday’s elections; Boehner, the speaker-in-waiting, seemed intent not to gloat.

Yet the clearer reality is that these are men of vastly different agendas, styles and backgrounds.

And it was telling that just about every mention of cooperation between them was accompanied by insistence on more give by the other — essentially the same formula for bitter gridlock that existed before voters tilted power toward Republicans.

“The new majority here in Congress will be the voice of the American people,” declared Boehner, whose mission includes undoing Obama’s signature health care law.

Obama offered an opposite analysis, saying any mandate to debate and vote again on the issues of the past two years would be “misreading the election.”

These are men who simply see solutions to problems differently. Likewise, their ways of going about their business.

There’s really no connection between them when they do talk.

That’s how Boehner bluntly put it before the election, and the White House does not dispute the feeling. Obama is known to poke fun at Boehner’s perpetual tan, and they both enjoy a good round of golf, but Obama surely has it right when he says the finding of common ground will not be easy.

Boehner is an amiable political animal, a happy warrior who came of age on Capitol Hill during the messy years of the so-called Republican revolution under former Speaker Newt Gingrich. In those days, he reveled in hanging out just off the House floor, smoking, chatting and collecting intelligence from almost anyone who ambled by.

Boehner is a backslapper with a sarcastic wit and a penchant for getting worked up, often choking up during floor speeches or losing his temper altogether.

Obama is the Ivy League-educated law professor who is known for keeping his composure and publicly yielding few flashes of anger. When he was in the Senate, Obama stayed above much of the political back-and-forth. And with both houses of Congress controlled by Democrats in the first half of his term, Obama didn’t need much help, if any, from Republicans to pass his signature policy initiatives.

Boehner was a prime Obama target during the lead-up to the midterm elections, with the president criticizing the Ohio congressman by name and setting him up as the embodiment of unwise Republican ideas, past and future.

The White House went so far as to choose Cleveland as the site of an early September speech on the economy because Boehner had delivered an economic address in the same city two weeks prior. The president called out Boehner eight times.

And the two men have been involved in some bitter face-to-face exchanges, such as a classic White House meeting in 2008 about the financial bailout.

Boehner bluntly aired the House GOP’s growing concerns over the plan, while Obama — then a presidential candidate — said some lawmakers simply didn’t understand the urgency of the situation, something Republicans interpreted as a swipe at them.

Obama did call to congratulate Boehner Tuesday night, but he made only a vague reference to looking forward to working together and to meeting in the next few weeks. There is no expectation of a meeting between the men before Obama leaves on Friday for a 10-day trip to Asia.

Republicans regard Obama as haughty and unwilling to engage; Boehner himself accused the president earlier this year of offering “finger-wagging lectures” instead of leadership.

And Obama and Boehner are not believed to have ever met one-on-one, with their dealings conducted in group meetings or through senior aides.