Notice how all countries are shown, even though only half the globe is visible?
In fact, the whole world has first been transformed conformally onto each hemisphere of a globe, using a construction by Edgar N. Gilbert, resulting in a globe with two images of the world, one on each hemisphere. It is then projected to 2D using the orthographic projection.
Gilbert had a real globe like this to play tricks on unsuspecting visitors:
When people visit Gilbert’s office, he likes to ask them what is wrong with his globe. If the visitor cannot see what is wrong, Gilbert gives the globe one slow, complete turn. “Even this hint”, he writes, “does not always succeed”.
–Martin Gardner, On Mathematical Games, Scientific American, Nov. 1975.
The Gilbert Two-World Perspective Projection is a useful visual illusion. It resembles the world as people will increasingly see it, from space and in the round. Also, the map communicates its message to the reader in a natural manner, by relying on each viewer’s well-developed powers of perspective sight.
–Alan A. DeLucia and John P. Snyder, An Innovative World Map Projection, The American Cartographer, Vol. 13, No. 2, 1986, pp. 165-167.