An Earth where the Eastern and Western emisphere are equal

Plateau-Madou Projection

Examples of Twinned Areas:

Australia = Greenland

Africa = South America

Europe = Central America

Asia = North America

Madagascar = Antarctica Peninsula 

New Zealand = United Kingdom

Tasmania = Iceland

Fenno-Scandinavia = Baja California Peninsula

Turkey = Florida

Italy = Yucatan Peninsula

Greece = Mississipi River Delta

Borneo = Baffin Island

New Guinea = Ellesmere Island

Japan = Victoria + Banks Islands

Philippines = Sverdrup Islands

Sumatra = Newfoundland

Sicily + Sardinia + Corsica = Cuba + Hispaniola + Jamaica

Strait of Gibraltar = Panama Canal

Tyrrhenian Sea = Caribbean sea

Adriatic Sea = Gulf of Mexico

Baltic Sea = Gulf of California

...

Here are a few curiosities (when you rotate the "hemisphere"):

1) Washington DC can be found in Iran!

2) In this map the term "American-Indians" would be justified...rather than an explorer's mistake :)

3) "Afro-americans" could be the Africans living in Brazil...rather than in the US

4)...