Head to Head: JFK and RFK, Los Angeles, July 1960

In July 1960, Democratic delegates from around America gathered in Los Angeles to nominate the party’s candidate for president. While a number of experienced and respected Democrats had their hats in the ring — including Missouri’s Stuart Symington, Hubert Humphrey, Adlai Stevenson (the party’s nominee in ’52 and ’56) and Lyndon Johnson — ultimately it was the young senator from Massachusetts, John F. Kennedy, who won the nomination on the first ballot, and went on to beat Richard Nixon, by the very slimmest of margins, in the general election.

Kennedy’s aggressive, efficient campaign leading up to Los Angeles was led by his brother, Robert, and most of the old guard of the Democratic party was caught off-balance by the energy and the charisma that JFK exuded on the stump. He was, as many commentators have pointed out, greeted as something of a rock star wherever he went; the other candidates, to their great misfortune, were treated as mere politicians.

That John Kennedy had a beautiful, sophisticated wife by his side, a young daughter at his knee and another child on the way only added to the image of a dad who could very likely relate to the concerns and hopes of young American families of every political persuasion.

Here, LIFE.com remembers Hank Walker’s famous photograph of JFK and RFK conferring in a Los Angeles hotel suite during the 1960 Democratic convention — a photograph that speaks volumes about the bond between these two intensely ambitious and pragmatic brothers. In fact, Walker made his picture at the very moment when that brotherly bond and the vaunted Kennedy pragmatism clashed head-on.

In John Loengard’s excellent book, Life Photographers: What They Saw, Walker described the scene playing out in front of his lens — a scene far more fraught than a cursory glimpse at the photo might suggest:

All these years later, knowing the awful fate in store for both of these complicated men, this quiet moment — shared, through Walker’s tough, sensitive artistry, with the rest of us — feels like the end of something.

At the time it was made, though, Walker’s evocative photograph probably felt like the very beginning.