With the Republican presidential primary contest essentially over, now begins one of Washington’s great mating rituals: the Dance of the Possible Vice Presidents. In which politicians who want the country’s second-highest office of trust must first prove that they can lie about it.

“I’ve made it clear I don’t want” the job, Sen. Joseph R. Biden Jr. said in 2008. “I’m satisfied where I am right now.”

“I have absolutely no desire to go back to government,” Richard B. Cheney, then an executive at an oil-field company, said in 2000.

Both men later accepted the vice-presidential nod they had said they didn’t want. In Cheney’s case, he led the search that eventually led to . . . him.

This is expected. The first unwritten notion about the “veepstakes” is that the American vice presidency is like a bronze medal or a day-old doughnut: Sure, you might enjoy having it. But it is considered unseemly to want it.

The dance is expected to begin in earnest now that Mitt Romney is the de facto Republican nominee. On Monday, the candidate said he has named Beth Myers, a longtime senior adviser, to lead his running-mate search committee, even though he acknowledged that it is “way too early to begin narrowing down who the potential vice-presidential nominees might be.”

The selection of a running mate is a vitally important political calculation for any presidential nominee: It can help swing a key state or voting demographic or reassure voters that a capable second is ready to take over in an emergency.

And in most cases, it is the first big executive decision made under the kind of public scrutiny that presidents face every day. For that reason, it is crucial not to crowd the decider — or to seem like you’re stealing his airtime.

“You’re number two, not number one,” said former vice president Dan Quayle — a politician who, in his day, was considered to have carried out the rituals of the office with rare skill. “And if you’re out there actively campaigning, there’s a subliminal message: that maybe you won’t be that comfortable in the number-two position.”

These odd rules of the vice-presidential courtship have not changed, even as the job itself has. In the past two White Houses, Cheney and Biden have played significant roles in policymaking and politicking — a major break from the old days, when their office was likened to a spare tire.

Or “a warm bucket of spit,” a description attributed to Depression-era vice president John Nance Garner.

“It makes you one of the most significant people in the country,” said Joel Goldstein, a professor at Saint Louis University who studies the vice presidency. “There’s no embarrassment in wanting to be secretary of state, or secretary of the Treasury, or secretary of defense. There shouldn’t be in wanting to be vice president, either.”

This year, most of the top contenders have been acting out the right steps. Most of them.

Every four years, a few have to learn — the hard way — that the last thing a vice-presidential hopeful should show is hope.

“One name to keep in mind would be my boss, Rep. [Cathy] McMorris Rodgers,” an aide to the congresswoman from Washington state wrote to reporters this month, talking up his boss as a potential running mate for Romney. And violating several rules of the game.

This discussion is likely to occupy Washington for the next few months, filling a gossip gap between the primaries and the height of the general-election campaign. If Romney follows tradition, he won’t choose until summer.

Until then, these are some things to keep in mind:

1. You are focused on your day job.

“I’m busy trying to do my job in Congress,” House Budget Committee Chairman Paul Ryan (R-Wis.) said on NBC’s “Today” show recently. “So, if we have to cross that bridge, I’ll make a decision then, but I haven’t given it the serious kind of thought, you know, with my family, to give an answer.”

This is the Old Reliable of vice presidential dodges, hallowed by usage and time. This year, it also has been used by Sen. Rob Portman (Ohio), who is mentioned as a top contender: “I love representing Ohio. . . . I just went through about a 20-month election campaign to get the honor to do that, so it’s the best place for me.”

The beauty of this tactic is that it provides its own escape hatch. What if, while a politician was focusing so deeply on his or her daily work, Romney sidled up undetected and popped the big question?

“I owe it to Governor Romney, if he were to ask me the question, to sit and listen to him as to why he thinks I would be the best person to be vice president,” said New Jersey Gov. Chris Christie, another possible running mate. “From my perspective, if you’re a betting person, bet on me still being governor of New Jersey in January of 2013.”

2. You won’t get it. In fact, you don’t even want it.

As a potential vice president for Romney, Sen. Marco Rubio (Fla.) would appeal to tea partyers and Latinos. Rubio is an energetic campaigner with an inspiring life story. Former Arkansas governor Mike Huckabee said recently that the first-term senator would be Romney’s strongest choice.

Rubio, following the ritual, has professed not to want the job.

“I’m not going to be vice president,” he told the Tampa Tribune this month. “I’m not going to be asked to be vice president. I don’t want to be vice president.”

He repeated the mantra again in an interview with CNN en Español, saying: “I’m not going to be the vice president.”

Of course, that’s what Vice Presidents Cheney and Biden also said.

Here’s where the veepstakes gets confusing. Because its rituals encourage all contenders to say no, it can be difficult to discern when one of them actually means it.

This year, for instance, New Mexico Gov. Susana Martinez has been repeatedly pressed about the post — despite the fact that she has always said she’s not interested.

The reason, Martinez says, is that she is the guardian for her developmentally disabled sister. A move to Washington, the governor said, would be “devastating” for her family.

“I just couldn’t do it,” she said.

Nevada Gov. Brian Sandoval, another prominent Hispanic Republican, has also deflected the idea that he could be Romney’s running mate. He has said he plans to run for reelection in 2014 and has suggested Rubio, Martinez and Virginia Gov. Robert F. McDonnell as candidates for the job.

3. Keeping silent is acceptable. Barely.

It is not against the rules of the veepstakes to be coy when asked about the possibility of becoming vice president. But it certainly is dangerous.

A contender’s silence is a signal of the forbidden emotion: the natural human excitement that attaches itself to the thought of holding the United States’ second-highest office.

“You have to ask him,” McDonnell told Real Clear Politics in January, motioning toward Romney at a campaign event. Then, perhaps realizing that he had already said too much, the governor reverted to rule No. 1: “I’ve got the job held by Thomas Jefferson and Patrick Henry — there’s not a better job in America.”

Old political hands say that this kind of tactic runs the risk of looking overly forward.

“It’s the clearest signal that is given before the election about who this [would-be] president really is,” said former vice president Walter Mondale (D). He went through the process both as a selectee and a selector, when he picked Rep. Geraldine Ferraro as his running mate in the 1984 campaign. “I don’t think the [candidate] should be crowded.”

There’s also a worry of ruining the surprise.

In 2004, for instance, Sen. John Edwards (N.C.) had long been rumored as a potential running mate for Sen. John F. Kerry (Mass.), the Democratic nominee — and Edwards did little to dispel that idea. “I’m going to remain silent,” he said. When Edwards was chosen, Kerry got little help in the polls.

4. There is a place to campaign. It’s just not in public.

For vice-presidential hopefuls, public silence should mask a backroom charm offensive. They should demonstrate their ability to raise money — for their party’s nominee, or for themselves. They should campaign for the nominee, if asked.

And they should keep the candidate’s secrets.

“I told a couple people on my staff, and I really didn’t tell anybody else,” Quayle recalled, after Republican nominee George H.W. Bush made an initial inquiry about the vice-presidential slot in 1988. Other nominees’ names leaked. But Quayle’s didn’t, and he considered that a test of loyalty. “I wanted him to know,” Quayle said. “That’s what he wanted, and I wasn’t going to” leak.

What they shouldn’t do is say any of this aloud.

That’s what made the e-mail from McMorris Rodgers’s office so unusual: not that she might want the job, but that she might be willing to admit it.

“I think a very good case can be made for including her in a future story about the VP contenders,” her aide wrote to reporters.

McMorris Rodgers certainly has things to recommend her: a four-term congresswoman, she is the highest-ranking woman in the House GOP leadership. In a recent interview, the aide said the idea had not come from McMorris Rodgers. In fact, she had nothing to do with it. But what was it based on?

Had the Romney campaign actually made inquiries about choosing her as a vice presidential nominee? Not so far, the aide said: “It’s never come up.”

Staff writer Ed O’Keefe contributed to this report.