If this week’s House Republican leadership elections told us anything, it’s that we should put to bed this tired meme about a civil war between the tea party and establishment Republicans.

On June 10, when House Majority Leader Eric Cantor (R-Va.) was toppled by a libertarian challenger in the Republican primary, the word went forth throughout the land that the tea party was resurgent and that House Republicans would tack even more to the right.

Nine days later, Cantor’s colleagues held an election to replace him, and the winner was California’s Kevin McCarthy — who has a voting record more liberal than Cantor’s. McCarthy became majority leader by clobbering a tea party challenger, Raúl Labrador (Idaho). A more reliable conservative, Steve Scalise (La.) won the No. 3 leadership slot by beating not only a more moderate challenger but also a tea party opponent who regarded him as too cozy with the establishment.

How to explain the contradiction? Simple: Ideology had little to do with the elections.

For all the talk of the struggle between the tea party and the GOP establishment, it has become a false dichotomy. In ideological terms, the tea party has already won the battle; Republican lawmakers new and old have shifted so far to the right that the differences among them are minor. What divides Republicans now is temperament — and here the establishment has prevailed.

The real split among congressional Republicans is between the bomb-throwers and the legislators. On Thursday, the bomb-throwers lost badly. Those who followed the old-fashioned rules of politics — building relationships, trading favors, balancing regional interests — prevailed.

That’s how to understand why McCarthy, with his 72 percent conservative rating, trounced the 100 percent Labrador. In an interview with my Post colleague Robert Costa earlier in the week, Labrador said of McCarthy: “I hear he does know the names of spouses, which I guess is a big issue. But you know, this shouldn’t be about personalities.”

As Labrador learned Thursday, it is all about personalities.

This doesn’t mean politics will return to normal functioning anytime soon. With Scalise (chairman of the conservative Republican Study Committee) in the leadership, House Speaker John Boehner’s caucus could become even less governable. But the overall message is hopeful in the long term. As Labrador tried to round up votes for his candidacy, he heard concerns about his temperament. On the hustings, anger works. But in the legislature, conservatives are wearying of bombast.

“If the job of the majority leader were to get everybody to vote no — well, then the other guy would’ve won the race,” Rep. Charlie Dent (Pa.), a (relatively) moderate Republican. “Some of our members with the harder edges have a harder time getting to yes. The perfect’s always the enemy of the good, incremental progress is tantamount to capitulation. . . . It’s harder for them to field a candidate who can get folks to support an agenda.”

The professional right was duly outraged by Thursday’s results. Longtime activist Richard Viguerie called McCarthy and Scalise “go-along-to-get-along Washington insiders.” Sen. Ted Cruz (R-Tex.) invited the vanquished House hard-liners over for pizza.

Exactly how few votes there were for Labrador’s just-say-no position is unclear; Labrador, on his way to defeat, proposed that McCarthy’s election be unanimous. But Rep. Justin Amash, a prominent Labrador backer, made no attempt to hide his disappointment. “People told us they wanted a significant change in our leadership; they wanted us to pursue a more conservative agenda, and I don’t think grass-roots Republicans are going to be satisfied with the outcome,” the Michigan tea partyer said, grumbling about chairmanships and other favors bestowed by leaders to win support.

Messages from voters weren’t as important to lawmakers as the stylistic differences among the candidates — what one Republican called the “relational” considerations. Labrador failed to grasp that. He arrived for Thursday’s vote without an entourage and slipped in through a back door marked “private.”

McCarthy, by contrast, stood in the front of the room before the vote with committee chairmen and other high-powered friends; in his victory remarks, he called himself “a guy that spent his time going around recruiting many of these individuals to get the majority.”

In the race for the No. 3 spot, the majority whip, Scalise, a well-liked figure among colleagues, demonstrated his support by leading a procession of more than 50 supporters into the meeting; after his victory, he spoke of what he had done “to build consensus.”

Yes, the tea party has pushed Republican ideology far to the right, but the establishment way of politics — the horse trading, logrolling and backslapping — has prevailed, and it always will. It’s human nature.

Twitter: @Milbank

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