On 17 May 1969 Leonard Bernstein ended his 12-year run as musical director of the New York Philharmonic with a performance of Mahler’s Third. The next night he went to see Jimi Hendrix play Madison Square Gardens. And there you have him. Was Bernstein a fragile romantic or a firebrand rocker? Was he the spiritual visionary who gave us Chichester Psalms or the tin-pan-alley tunesmith behind West Side Story?

Bernstein went to his grave claiming it was possible to be all these things and more — insisting that you could be a political activist and a concert pianist, a conductor of...