That word "classical," as applied to music, is a source of endless confusion. It is common to speak of "classical" music as the antithesis of ''popular'' music without once considering that most of it was originally intended for popular consumption. I suppose that we partly , confuse it with "classic" which refers to supernal excellence: classics are built to last. The terms both come from Latin classicus, which denotes ancient Romans of the best families, and itself derives from classis, which means simply a class of people or things). But "classic" also means ''like the arts of the ancient Greeks and Romans." This would seem inapplicable to music, since we know next to nothing of the music of classic times. But by extension, it can mean "ordered, logical, balanced, formal, etc.", and so "classic" is applied to the music of the era of Mozart and Haydn to distinguish it from the subsequent romantic music, which is, allegedly, anything but ordered, logical, balanced, etc., etc.
And this is where neo-classical music fits in. Supposedly it represented a protest against the let-it-all-hang-out emotionalism of the Romantics -- particularly, I suppose, such late Romantics as Tchaikovsky and Mahler and their ilk, who musically-speaking, stripped themselves naked and writhed in the bulrushes. Its motto was, in effect, "cool it," and it cooled it by attenuating, clarifying, and applying strict formal principles. But curiously, however "neo" it may have considered itself, it turned not so much to the Classic Era as to the Baroque. Its chief deity was not Mozart but Bach. At the same time, it maintained its modernism through a use of ragged, asymmetrical tunes and piquant and "shocking" dissonances timed to go off where one least expected them. (It is an interesting fact that whereas the neo-classicists and the serialists saw themselves as natural enemies, their chief stock in trade was formalism -- which is no doubt one reason that the leading neo-classicist, Stravinsky, became in his last years a leading serialist.)
Who began neo-classicism is probably a futile consideration. Some attribute it to Ferruccio Busoni in the early years of the century; others like to date it from Stravinsky's wind octet U923). Stravinsky himself, however, had no doubts about it: he tells us that he wrote a polka for piano four-hands which was intended to caricature Diaghilev both as impresario and pianist. He played it for its subject in a Milan hotel room one day in 1915, with the composer Alfred Casella in attendance. Says Stravinsky, it opened up a new path to Casella; "so-called 'neo-classicism' ... was born in that moment." Heigh-ho! Well, anyway, Stravinsky added a march in homage to Casella, a valse for Erik Satie, and later five more pieces for his young son and daughter to play. Eventually he orchestrated the eight into the two delightful and witty and rarely-heard orchestral suites performed here.
The "Dumbarton Oaks" concerto, which is heavily indebted to Bach, including a theme from the third Brandenburg, was commissioned by Robert Woods Bliss, who lived in the magnificent estate of that name in the Georgetown section of Washington, D.C., in which area I was residing at the time. Since I walked past Dumbarton Oaks almost every day, I remember thinking in my innocence "How pedestrian to name a work after something so commonplace!" (I was used to names like "Eroica" and "Pathetique" .)
The ballet "The Card Game" (original choreography by Balanchine) was completed in 1936. It is in "three deals" and, if I recall correctly, the dancers represent cards rather than players. Written in a period when Stravinsky was playing footsie with the Romantics (here Rossini, Bellini, and Donizetti) it seems temporarily to have fallen out of favor, though I've always enjoyed it.