Saul Steinberg’s take on the world had a way of being spot-on.

For the artist, wrinkled senior citizens were synonymous with “Florida types”; erotica had something to do with an intertwined question mark and the numeral five; and the otherwise square state of Wyoming could be compared to the sum of the states, countries and regions (Switzerland and Albania among them) that add up to its 97,914-square-mile area.

And yet somehow it makes sense.

The touring retrospective “Saul Steinberg: Illuminations,” which recently opened at the Smithsonian American Art Museum, is an ideal introduction to the taxonomy of Saul. It’s packed with his famous New Yorker covers and cartoons and also plenty of stuff not readily associated with the late artist, such as Dadaist collages, letter sculptures and comic book-style surrealism.

Steinberg, an illustrator and cartoonist who fittingly dubbed himself a “writer who draws,” was a master of the satiric classification system, an affectionate mocker of prototypes and mind-sets who managed to pull off his incisive assessments with a measure of playfulness that makes his art — whether drawings, cartoons, paintings or sculpture — impossible not to enjoy.

See those well-heeled New York society ladies with their feathers and enormous fur coats, those important people who end up on pedestals, those fou-fou 19th-century denizens art gazing at an overcrowded salon show? Absurd, Steinberg seems to be saying, but not altogether pernicious.

Meanwhile, the cowboy (a stock character in American life) is revealed in his most ridiculous state surrounded by his felled victims, while staples of Middle American commerce — Sears Roebuck, the Sunoco station, an A&P grocery — peep out on the horizon. The national pastime, however, gets more hagiographic treatment. Brooklyn’s old Ebbets Field at night is cast in an almost divine light, with the baseball park emerging in the drawing as the only hopeful beacon in an otherwise murky and faintly sinister landscape.

The psychological underpinnings of post-war American life are a constant Steinberg theme. His “Man in the Gray Flannel Suit”-invoking “Group Photo” (1953) features a series of suited men composed of nothing more than thumbprints. The only distinction: an arrow pointing to the head of one of these yes-men, which, ironically, is rubber-stamped in place. In another image from that same year — “Techniques at a Party” — the guests’ internal character is manifest in their aesthetic. It’s a hilarious jape, in which an otherwise bourgeoisie cocktail party takes a distinctly unorthodox turn featuring a big-headed princess, a pointillist woman, a clown and a man so undefined he appears to be disappearing into thin air.

But it’s not just the mind-set of post-war life Steinberg explored — it also was what Americans were doing to their environment. In “Motels and Highway” (1959), a superhighway jammed with bug-like cars cuts through a landscape filled with signs for motels and diners. From the roadside, Steinberg’s grinning croc surveys the scene. It was a view of an America, poet Stephen Spender would write, “acknowledging no debts to the past and no need to fit its bursting ubiquitous forms into the surrounding scenery which it tramples down to suit its purpose.”

And Steinberg perfectly plumbed the essence of all sorts of places. He filled an ink drawing of Venice’s Piazza San Marco — its detailed arcades rendered in Steinberg’s architect-trained hand — with a mishmash of pontificating Italians, pigeons and camera-happy, sunglasses-wearing American tourists. Here, the oversized birds are represented out of proportion to their human counterparts, while the bulbous basilica shrinks under the shadow of the looming bell tower. Then there’s the kooky 1970s Greenwich Village he portrays in his “Bleecker Street” — where many of the characters look like they wandered off the set of “Star Wars” — and a Los Angeles landscape, where the neighborhoods, traffic and trees resemble the amorphous shapes of a Henry Moore sculpture.

Some of Steinberg’s most fetching (and famous) work had him playing the part of a Manhattan-centric New Yorker, whose inability to see much beyond the confines of the Hudson and the East rivers is behind both the silliness and the greatness of the images. His widely cribbed “View of the World from 9th Avenue” (1975), with its solipsistic perspective, loses all definition once you cross the Hudson. From there on out, according to Steinberg’s Manhattanite, the country’s basically just one big green prairie till you get to the Pacific. Then it’s just a hop, skip and a jump from there to China, Japan and Russia. (Likewise, in 1973’s “The West Side,” he shows the rest of the United States as a remote, and blob-like, borough of Manhattan.)

In reality, Steinberg, a Romanian-born Russian Jew who fled from Italy to the United States to escape fascism, was anything but a parochial Manhattanite. (He privately chastised the thinking some of these pictures represented as that of “the crummy people.”) After military service in World War II, he spent a good deal of his life on the great American road and traveling the world at large. (One of the more mind-bending pieces in the show is a 29-section drawing, created for a mural, aptly titled “The Line” (1954), which demonstrates the chameleon-like possibilities of his craft in a mere 33 feet.)

The exhibit offers plenty not normally associated with Steinberg’s New Yorker output.

For instance, Steinberg merged photography and drawing, taking a black and white print of a bathtub filled with water and adding the outlines of a woman stretched out in the tub. Likewise, a shot of freshly excavated, underground piping is augmented by drawing in a tiny, cliff-side village. He also used cheap tourist postcards (known as “chromes”) and imprinted them with a rubber stamp of some praying peasants from a canvas by Millet — in the process injecting them into all sorts of banal backdrops. These simple 19th-century country folk end up at the railroad station in New London, Conn., on a modern strip in Miami Beach, Fla., and at the beach. Steinberg, the immigrant, also played with the notion of passports and other “official” documents, which he created and adorned with meaningless handwriting. Then there were his Monty Python-style dada works — such as an image of Don Quixote jousting at a pineapple or another filled with legions of rubber-stamped artists at easels, cyclists and marauding Indians.

Steinberg’s whimsy extended beyond the page. He made quirky brown-bag masks (in which he often was photographed) and wooden books and even created a life-size desk filled with volumes ranging from Stendhal to a Sotheby’s catalog and rendered as if it were the crude cartoon sketch of a desk come to life from one of his own illustrations.

The version of the show on view at the museum also features a series of sketches he created using Smithsonian Institution letterhead when he served as its first (and last) artist-in-residence in 1967. Playing off the tiny insignia of the Smithsonian Castle at the top of the stationery, Steinberg included the decoration in a teapot, the monogram on a man’s French cuff, the T-shirt on a bottomless female form and a baronial country view. According to the Washingtonian story “Saul Steinberg: An $11,000 Joke on the Smithsonian,” Steinberg completed only four months of what should have been a yearlong residency before returning to New York without much to show for his time or the institution’s money.

Good to see there were no hard feelings on the Smithsonian’s part.

“Saul Steinberg: Illuminations” is on view through June 24 at the Smithsonian American Art Museum, located at Eighth and G streets Northwest. For more information, go to americanart.si.edu.