"The elder brother is the natural rival," wrote Sigmund Freud in a letter to Thomas Mann. "The younger one feels for him an elemental, unfathomably deep hostility for which in later life the expressions 'death wish' and 'murderous intent' may be found appropriate." This may not explain why Ed Miliband stood against his brother for the Labour leadership in 2010, thus unleashing a very public psychodrama whose latest act opened yesterday with the elder brother putting the Atlantic Ocean between himself and his younger powerful rival, but it does explain other cases of sibling rivalry.

Think Cain and Abel, Romulus and Remus, Noel and Liam. "That's why we'll be the best band in the world, because I fucking hate that twat there," reflected Liam philosophically after storming out of a Heathrow departure lounge when Oasis were in their mid-90s heyday. "I fucking hate him. And I hope one day there's a release where I can smash fuck out of him, with a Rickenbacker, right on his nose."

I know what you're thinking. Why must an innocent Rickenbacker suffer? But Liam's enduring guitar-fixated death wish does point up a singular truth that Cain and Romulus knew: sibling rivalry isn't over until one party dies. Sure, Freud maintains that Napoleon and brother Joseph had a warmer relationship later in life and that crushing Europe was the former's projection of his erstwhile loathing for his elder brother, but that seems fanciful.

Frank admissions of loathing are always more edifying than PR guff for the credulous about brotherly love.

When David declared his candidacy for the Labour leadership election, he announced that "brotherly love is more important than politics", trumping Ed's declaration that David is "my best friend in all the world. I love him dearly". Even Hallmark cards would spike those sentiments.

Yesterday, there was more of this intolerable treacle. "As for us," said Ed of his relationship with David, "we went through a difficult leadership contest but time has helped to heal that. I will miss him. But although he is moving to America, I know he will always be there to offer support and advice when I need it." If that wasn't enough, David had to put up with being biffed with the tainted stick of praise, in the form of an encomium from Tony Blair. If only Ed had added: "But, incidentally, I will never need it."

But at least we got from one Miliband an admission yesterday that there had been wounds inflicted on the putatively non-feuding brothers by the leadership contest, even if we were only to learn that they existed retrospectively, long after they had been covered up and even if the admission was couched in the cliche of time, the great healer. The gap between what happened and what's said in political-speak is usually huge, but eminently deniable – so we must be grateful to Ed for admitting there was such a gap. Even if it has long been filled in, patted down, grassed over, planted with trees and comprehensively repressed. Even if all the principal protagonists have moved on, so that we won't need to think about what happened now that we're going forward, thanks very much. This press conference is over.

But hold on. Why was the leadership contest difficult if, indeed, brotherly love transcends politics? Because, surely, that treacly sentiment is a lie: no sibling relationship would emerge psychically unscathed from such a contest as the one that David and Ed fought.

Dorothy Rowe, the author of My Dearest Enemy, My Dangerous Friend: Making and Breaking Sibling Bonds, told me three years ago: "Most of us like to be seen to behave well, even if in private we're not. If you lead a public life, you will have developed lots of strategies for not letting your real resentments show." And the Milibands' past three years in public life have demonstrated their mastery of those strategies – even if we don't really believe neither has felt real resentment for the other during that time.

It's understandable that David cites one reason for his flight to head up a charity in New York as being to get away from forever being judged through the "brother prism".

But even by going so far away from his brother he enacts a venerable move in the sibling rivalry. In Marina Lewycka's A Short History of Tractors in Ukrainian, the narrator Nadia says to her hated big sister Vera: "See how we grew up in the same house, but lived in different countries?" Ed could now email David the same message.

In a prescient article for the Independent the day before the Milibands contested Labour's leadership election three years ago, Amy Raphael wrote: "If David should lose today's Labour leadership election to his younger brother Ed, he could suffer not just a political disappointment but also a profound feeling of annihilation." Raphael argued that it "will be better for Labour if David wins. A younger brother could work for an elder. I'm not sure if it would ever stick the other way around." And so it proved: David stepped back from the shadow front bench, turned from international politics to teaching at his old school, reversed from parliament and now backs away to another continent – if not to lick his wounds (there's a statute of limitations on wound-licking), then to master his feelings of annihilation.

But Raphael's point is important: there is an emotional asymmetry determined by David and Ed's ages. Had the young pretender failed, he could, easily, have worked for the elder shoo-in, but not the other way round. That's not to say, had Ed been defeated and yet served in David's shadow cabinet, everything would have been peaches and cream. After all, what some psychologists call the Remus complex would have been operative in that hypothetical scenario: namely, the resentment of the younger brother towards his more successful elder brother would have induced Ed to strive to outdo David.

But what we have been witnessing since Ed's surprise election victory is not so much Ed's Remus complex as David's Romulus complex, the horror felt by the elder sibling when the younger begins to succeed in his challenge. Poor Romulus: first he had to put up with baby Remus getting all the parental attention (the older child's eternal complaint that has kept shrinks in the money ever since Freud's consulting room opened for business) and the beast teat from his she-wolf of a mother, then he had to witness the little bugger rival him for the throne of Rome. So he killed him. Problem solved.

I ask Mary Beard, professor of classics at Cambridge University, whether Romulus and Remus offer any help in understanding the Milibands. "Basic point is that both can't be king," she says. "Romulus kills Remus – and that becomes a model for Roman civil war ever after. Brother always fights brother!"

Quite so. Cain slayed Abel for being more favoured by God than he. Romulus did in Remus because no one has invented the two-headed crown for brothers to share. Today things are different. David could not slay Ed for swinging the trade union bloc vote behind his candidacy (well, he could have, but that would have put quite the blot on his CV). Instead, he had to deal with his feelings of annihilation while continuing to serve as the representative of the electors of South Shields (tough gig). He had to live with his upset. And not just him: his wife, the violinist Louise Shackleton, was reportedly "absolutely furious" at the manner of his defeat and all-too-understandably sobbed from the sidelines.

There is even the story, which it is very hard not to be touched by, of the crestfallen David reciting to Louise the speech he had planned to deliver in the back of his car on the drive home from party conference. The final draft, leaked to the Guardian, was, it has to be said, impressive, stressing Labour's need to regain the public trust on the economy in the aftermath of the Brownite nonsense that Labour could abolish the business cycle. Just a shame he didn't get to deliver it to a bigger audience. It's hard not to imagine that painful journey, those dreams of what might have been replaced by the shocking realisation of what really transpired. Bad enough to be beaten, but beaten by a brother.

David Miliband addresses the media to announce his resignation as an MP. Photograph: Peter Macdiarmid/Getty Images Europe

Reading yesterday's announcement by David Miliband, it is hard not to interpret it as a belated return from that annihilation, an understandable and proud assertion of his power and importance after three years in which those have been systematically stripped from him.

Speaking about the International Rescue Committee (IRC), the charity to which he is going, he said: "The organisation was founded at the suggestion of Albert Einstein in the 1930s for those fleeing the Nazis, so given my own family history there is an additional personal motivation for me. I feel that in doing this job I will be repaying a personal debt." Quite so: as a Jew whose parents fled the Nazis, he has chosen to honour his family heritage and do noble, important work. Certainly, better act to change your destiny than do what Edward Heath did after being beaten in the Conservative leadership election of 1975 until his death 30 years later: sulk.

He added: "I will forever be Labour. But after writing two election manifestos in 1997 and 2001, and serving as a minister for eight years, I now have to make a choice about how to give full vent to my ideas and ideals." If you wanted to parse this in terms of the Romulus complex you could do worse than this: my brother may lead the Labour party, but I've done so much important stuff in that institution, now I'm moving on to a new challenge, something arguably more important than anything Labour's doing.

"I hope you will understand that the opportunity to lead the IRC represents a unique chance to put my experience into practice on behalf of some of the least fortunate people on Earth." It is a quite brilliant statement of a sidelined man's insistence that he is back as a figure to be reckoned with – as a player who has been marginalised too long. And why not? What sentient being embraces their symbolic annihilation by a sibling?

David Miliband hugs younger brother Ed after the latter won the Labour leadership battle. Photograph: Leon Neal/AFP/Getty Images

What gets lost here is the shock David must have felt at losing to Ed. "As small children, our sense of being a person is constantly under threat because we know so little of the world, we make so many mistakes, and other, older people can bamboozle us and use their power over us to humiliate and hurt us," argues Rowe. "Having your big brother steal and destroy the little shells you were treasuring can be more devastating than having your opera bomb or the critics slate your acting." Absolutely. But to grow up and find your little brother stealing your party from under your nose must be even worse. Specifically, it must be worse because you can't do what you would have done in similar circumstances as a child. You can't rage, scream, duff up your rival or wail to mummy – rather, you have to suck it up and say that you love the person who has crushed your ambitions and thwarted your manifest destiny. What hard work that must be, especially if the humiliation is so public!

None of this, incidentally, is to say that David doesn't love Ed and Ed love David. It's just that the truth of any sibling relationship must be more complicated than that. Even Christopher and Peter, the fabulous feuding Hitchens brothers who spent years savaging each other in print and ignoring each other in person, might have loved each other. But love isn't the whole story, especially where brothers are concerned. When the Guardian brokered a rapprochement in 2005 between the Hitchens brothers, Peter nailed the problem. There was an old joke in East Germany that went: "Are the Russians our friends or our brothers? And the answer is, they must be our brothers because you can choose your friends."