In public and off-camera, Harpo and Chico were difficult to recognize by their fans without their wigs and costumes, but it was almost impossible to recognize Groucho without his trademark eye-glasses, fake eyebrows and mustache. More Groucho: https://www.amazon.com/gp/search?ie=UTF8&tag=tra0c7-20&linkCode=ur2&linkId=f120ebf09140f991af662ca585bd141f&camp=1789&creative=9325&index=dvd&keywords=groucho
The greasepaint mustache and eyebrows originated spontaneously prior to a vaudeville performance in the early 1920s when he did not have time to apply the pasted-on mustache he had been using (or, according to his autobiography, simply did not enjoy the removal of the mustache every night because of the effects of tearing an adhesive bandage off the same patch of skin every night). After applying the greasepaint mustache, a quick glance in the mirror revealed his natural hair eyebrows were too undertoned and did not match the rest of his face, so Marx added the greasepaint to his eyebrows and headed for the stage. The absurdity of the greasepaint was never discussed on-screen, but in a famous scene in Duck Soup, where both Chicolini (Chico) and Pinky (Harpo) disguise themselves as Groucho, they are briefly seen applying the greasepaint, implicitly answering any question a viewer might have had about where he got his mustache and eyebrows.
Marx was asked to apply the greasepaint mustache once more for You Bet Your Life when it came to television, but he refused, opting instead to grow a real one, which he wore for the rest of his life. By this time, his eyesight had weakened enough for him actually to need corrective lenses; before then, his eye-glasses had merely been a stage prop. He debuted this new, and now much-older, appearance in Love Happy, the Marx Brothers's last film as a comedy team.
He did paint the old character mustache over his real one on a few rare performing occasions, including a TV sketch with Jackie Gleason on the latter's variety show in the 1960s (in which they performed a variation on the song "Mister Gallagher and Mister Shean," co-written by Marx's uncle Al Shean) and the 1968 Otto Preminger film Skidoo. In his late 70s at the time, Marx remarked on his appearance: "I looked like I was embalmed." He played a mob boss called "God" and, according to Marx, "both my performance and the film were God-awful!".
The exaggerated walk, with one hand on the small of his back and his torso bent almost 90 degrees at the waist was a parody of a fad from the 1880s and 1890s. Fashionable young men of the upper classes would affect a walk with their right hand held fast to the base of their spines, and with a slight lean forward at the waist and a very slight twist toward the right with the left shoulder, allowing the left hand to swing free with the gait. Edmund Morris, in his biography The Rise of Theodore Roosevelt, describes a young Roosevelt, newly elected to the State Assembly, walking into the House Chamber for the first time in this trendy, affected gait, somewhat to the amusement of the older and more rural Members who were present. Groucho exaggerated this fad to a marked degree, and the comedy effect was enhanced by how out of date the fashion was by the 1940s and 1950s.