When John Cheever's editor suggested an omnibus collection of his short stories, the writer was nonplussed. "Why do you want to do that?" he asked. "All those stories have already been published." The resulting book, The Stories of John Cheever (1978), became one of the bestselling short story collections of all time, winning its author a National Book award and a Pulitzer. But it didn't just buttress Cheever's uncertain reputation. As Robert Morace notes, it "revived interest in the short story on the part of publishers and readers, making it both commercially more viable and critically more respectable".

Cheever published 121 stories in the New Yorker between 1935 and 1981, and others in Esquire, Playboy and elsewhere. His collected stories runs to nearly 900 pages, despite its exclusion, at Cheever's insistence, of his entire first collection. Hemingway's cadences, clearly discernible in the early stories, slip from view around the time of The Sutton Place Story (1946), but much later, in The World of Apples (1966), the poet Asa Bascomb recites his personal pantheon before a Christian relic, ending with: "'God bless William Faulkner, Scott Fitzgerald and especially Ernest Hemingway.'" Throw in Chekhov and Flaubert, and Bascomb's list could be Cheever's. In addition to these, Elizabeth Hardwick notes, the "shadowy and troubled undergrowth of Cheever's stories brings to mind something of the temper of Melville and Hawthorne".

But, as Blake Bailey suggests in his extraordinary biography, Cheever absorbed his influences in such a way that identifying them adds little to the work. And if the workings of influence are obscure, in setting he really had no forebears: he became, by chance, the chronicler of Westchester, New York, just as it became the first large-scale suburban area in the world. His Shady Hill is a fictional territory to consider alongside Faulkner's Yoknapatawpha County or Thomas Hardy's Wessex. But while you might expect social commentary to trip across its manicured lawns, or satire to barge through the mushroom-coloured raincoats thronging its station platform, Cheever's suburban fantasias are a good deal stranger than that.

Proclaiming Cheever's strangeness should be a cliche, but for each generation the WASPiness of his settings seems to shroud his work in a phantom conservatism. Even if some of his most famous stories are fables about radios that transmit neighbours' private conversations (The Enormous Radio), or angels of death walking the streets of Manhattan (Torch Song), the casual observer's general impression remains one of cocktail hours, affairs and hungover Sundays. All those elements are there, of course, but rarely in ways one would expect. "[N]othing in Updike that isn't in Cheever first," writes Alan Bennett, but the similarities are in truth superficial; Updike is much less anomalous. Ben DeMott called Cheever's stories "dense in inexplicables", and Cheever himself told the Paris Review that fiction is "experimentation … One never puts down a sentence without the feeling that it has never been put down before in such a way … Acuteness of feeling and velocity have always seemed to me terribly important."

These words describe as well as any the atmospheres generated by Cheever's best stories, which include, I think, three of the best in the language. Each of these – Goodbye, My Brother (1950), The Country Husband (1954) and The Swimmer (1964) – also shares a profound duality. Cheever, as any account of his life explains, had duality at his core, and it veins the work. In Goodbye, My Brother, the scion of a New England family catalogues the faults of a disfavoured brother, Lawrence, before attacking him on a deserted beach. An early mentor found the narrative "troublingly uncertain". Cheever's response – "There was no brother; there was no Lawrence" – at once simplifies, complicates, and expands the story in several directions at once. The Swimmer is both the account of a man returning to his Shady Hill house via the pools of his friends and neighbours, and a dreamlike fable about a man stripped of everything. As with Kafka's Metamorphosis there isn't a single "true" reading – only two simultaneous realities.

Of these stories, though, it is The Country Husband (1954) that most perfectly accommodates this quality. It is, Edmund White tells us, the story that made Hemingway wake his wife in the middle of the night to read aloud to her. Here, everything is doubled. Surviving a crash-landing while on a business trip – the drama of which is completely ignored by his family – Francis Weed (the surname connoting both puniness and tenacity) returns to the idyll of home only to find a chaotic "battlefield". The babysitter is sick, and Francis falls in love with her young replacement. At a dinner party he recognises the French maid distributing drinks as a woman he saw stripped naked and spat on outside a Normandy village for collaborating with the Germans. The ambiguously named Shady Hill hovers between the end of summer and chilly autumn: "It was a frosty night when he got home. The air smelled sharply of change." I could go on; as Nabokov wrote, the story is "really a miniature novel beautifully traced, so that the impression of there being too many things happening in it is completely redeemed by the satisfying coherence of its thematic interlacings".

The story's justly famous closing lines take us away from a chastened Francis woodworking in his basement, and show us the curious evening rituals of Shady Hill:

"'Go home, Gertrude, go home,' Mrs Masterson says. 'I told you to go home an hour ago, Gertrude. It's way past your suppertime and your mother will be worried. Go home!' A door on the Babcocks' terrace flies open, and out comes Mrs Babcock without any clothes on, pursued by a naked husband … Over the terrace they go and in at the kitchen door, as passionate and handsome a nymph and satyr as you will find on any wall in Venice … The last to come is Jupiter. He prances through the tomato vines, holding in his generous mouth the remains of an evening slipper. Then it is dark; it is a night where kings in golden suits ride elephants over the mountains."

This widely quoted passage is exhibit A in the case for Cheever as, in Philip Roth's phrase, an "enchanted realist". But standing alone, this passage appears much more whimsical than it is. Nearly all its elements carefully reprise phrases and episodes from earlier in the story. Most importantly, that extraordinary final line echoes and transforms what a sentimental host tells the Weeds about his wife at the end of a party: "'She's my blue sky. After sixteen years, I still bite her shoulders. She makes me feel like Hannibal crossing the Alps.'" Reencountered here, is this mawkishness being reclaimed in an act of grace? Ironic mockery? Cheever's crowning achievement is the ability to be simultaneously generous and cynical, to see that the absurd and the profound can reside in the same moment, and to acknowledge both at the detriment of neither.

Next: Roberto Bolaño