When Igor Stravinsky’s poor old mother found that her hair was falling out, her famous composer son insisted that she pay for a wig out of her allowance. When he visited the Far East, the only thing he liked about the Japanese was that they didn’t expect tips in taxis or hotels. He could, in short, be fantastically stingy. Or generous,as the mood took him – particularly toward his family.

Stravinsky is that gift to biographers, a man of contradictions. In the first half of his Life of this quixotic genius, “Stravinsky: a Creative Spring,” Stephen Walsh described Stravinsky and Diaghilev raging over contracts by day, then getting sentimentally drunk by night, “like comic Russians.”

In “Stravinsky: the Second Exile,” the inconsistencies come thick and fast, and are often indistinguishable from plain hypocrisy.

Stravinsky could rarely be bothered to attend church, yet cluttered his houses with icons and relics; Mr. Walsh – never at a loss for a clever line – relishes the contrast between his music’s mathematical precision and “the religious superstitions of an Italian peasant.”

Politically, Stravinsky was a supporter of Mussolini’s fascists and the American democrats (though,sensibly,at different times). He welcomed his daughter’s marriage to a Jew, yet slipped naturally into country-club anti-Semitism. Perhaps this was because he had the nose and lips of a Der Sturmer caricature and was at pains to explain that, actually, he was a posh White Russian.

The contradictions in Stravinsky’s life, however, are as nothing compared with those in his music. The phrase “multiple personality disorder” comes to mind. Put it this way: If an ignorant but aurally acute person were presented with randomly shuffled CDs of the major works of almost any composer – Haydn, Beethoven, Mahler, Webern – he or she would be able to assemble them in roughly the right order.

But Stravinsky? Although the gloopy texture of “Firebird” clearly comes first, it would require fantastic acumen to guess the sequence of “Pulcinella,” “Apollo,” “The Rite of Spring,” “The Symphony of Psalms,” “The Soldier’s Tale,” “Les Noces,” “Petrushka,” and “Oedipus Rex.” His was “a notoriously unpredictable oeuvre,” as his biographer admits, which mimicked Italian baroque, French impressionism, Hindemith’s spiky counterpoint, Orthodox incantations, Russian folk song, and cabaret.

All this material, incidentally, was composed before 1934; so, in addition to defending Stravinsky against the charge of behaving like a musical cuckoo, Mr. Walsh has to confront the problem that the music produced in the second half of the composer’s life is historically less significant than his early masterpieces.

Yet this book is none the less readable for that. As an elderly resident of suburban Los Angeles, Stravinsky might not have provoked riots, but his reputation as “the world’s greatest composer” produced encounters with Walt Disney, Christopher Isherwood, Frank Sinatra, Aldous Huxley, Marlene Dietrich, Edward G Robinson, and, disastrously, Evelyn Waugh (“all music is positively painful to me”).

W.H. Auden, of course, wrote the libretto for “The Rake’s Progress,” one lateish work that has entered the repertoire of opera houses. There were also aborted plans for Stravinsky to write the score for a Charlie Chaplin Passion set in a drunken nightclub, a prospect too horrible to contemplate.

Mr. Walsh provides lovely vignettes from these years, several involving Stravinsky’s sturdy second wife, Vera. On one occasion, Mrs. Stravinsky made her way to the lavatory during a dinner party at the filthy apartment of Auden and Chester Kallman. Spotting a dish of brown liquid, she flushed it away – thus depriving the guests of Chester’s chocolate pudding.

Meanwhile, Stravinsky performed one last musical zigzag.The late scores might be neglected now, but in some respects they were as shocking as “The Rite of Spring.” Under the influence of his disciple Robert Craft (on the subject of whose truthfulness Walsh is guarded, Craft being still alive), Igor landed on the one nest he had sworn he would never touch: that of the Second Viennese School.The result was serialism of eye-watering astringency, some of it worked out on a pink piano in a cellar bar.

Mr. Walsh does not entirely refute the cuckoo charge. Stravinsky mastered more styles than any composer in history – who else could claim to have been a friend of Rimsky-Korsakov and Boulez? – and had something fascinating to say in each of them; but it is difficult to dispel the impression that his work adds up to less than the sum of its parts.

The same cannot be said of these two volumes, however, which convey complex musical analysis and jolly stories in masterly prose, and add up to one of the finest musical biographies of our age.