It’s always eye-opening to see the men and women behind famous works of art. And yet, while Picasso, Dali, Matisse and perhaps a handful of other “art superstars” are recognized around the world, countless other painters and sculptors of note — including some of the finest, most influential artists of the past century — are, in a sense, invisible. Their works endure; but their look — their faces, the way they held themselves — is often an utter blank in the mind of the casual art enthusiast.

Here, LIFE.com presents a series of portraits of the artists who re-imagined and redefined the way we see the world. Pollock is here, and Picasso, along with Georgia O’Keeffe and even Renoir. But can art aficionados actually identify a photograph of, say, Claes Oldenburg? Jasper Johns? Barbara Hepworth? Giorgio de Chirico?

[MORE: “Georgia O’Keeffe: Invincible”]

What’s wonderful about the pictures here, including those (like the deeply shadowed silhouettes of Hopper and O’Keeffe) in which we perceive a powerful sense of the full figure, rather than discrete details, is that each photo reveals something uniquely essential about the artist. One would be hard-pressed to find two more dissimilar portraits than those of Arshile Gorky and Robert Rauschenberg; but each of those captures an aspect of the artist’s personality and sensibility (Gorky’s moody intensity; Rauschenberg’s exuberance) that, in a heartbeat, helps us see something — even if it’s only a glimpse — of the human being behind the art.