|Portrait of FDR by Cecil Calvert Beall (c1933)|
Then or sometime later, it was apparently revised for promotional use in connection with FDR's residence at Warm Springs GA, describing it as "The Little White House," and introducing images of Eleanor Roosevelt and the United Nations.
Beall was a native of Montana who had studied at the Art Students League in New York with George Bridgman. It doesn't seem that "visual puns" were his usual way of working (look up Arcimboldo for historical precedents), but we do know that he used the same technique in at least one other painting. As shown below, it appeared in a wartime poster for the US Army Recruiting Service (presumably for World War I, if the soldier's helmet is correct).
In psychology, composite puzzle pictures such as these are typically referred to as "embedded figures." For a discussion of how these have been used in art, architecture and design (as well as, literally, in camouflage), here is an online essay.