The direction of the Sun is a dead zone in the astronomer's view of the Universe and the Solar System. This dead zone is not permanent because of Earth's rotation around the Sun, but it does makes some people wonder: could there be another planet in Earth's orbit exactly on the other side of the orbit, which is constantly hidden by the Sun?
Actually, the answer is more complicated. Since Earth's orbit around the sun is elliptic, and Earth's orbital speed is not constant, the Sun's visible disk is not big enough to cover a persistent spot in the other side of the orbit, so our twin would have been spotted already (according to my calculations, if the Sun was 7 times bigger, it would have effectively succeeded in hiding our twin)
So we easily ruled out the possibility of another planet on an identical orbit. But, there is another possibility: Earth's negative orbit. Take function p(t) as the vector of orbit in time t. -p(t) is also a valid planetary orbit (if you don't take inter-planetary forces into account). The sun is situated directly between -p(t) and p(t), thus it would effectively hide the two planets from each other. Unfortunately, the chance that another planet have assumed this negative orbit is much more slim.
The existence of such planet is more improbable as its gravity effects would have been detected by our measurements of the Solar System's celestial bodies in the same way Pluto was detected. So this planet, if it exists, must be not massive enough to be detected. So in the meanwhile, the negative orbit can be a useful hide-out place for extraterritorial space ships.
1969 movie: Journey to the Far Side of the Sun, by Gerry Anderson: A planet has been discovered which shares the same orbit as Earth but is on the opposite side of the sun. The European Space Exploration Council sends out American Glenn Ross and British John Kane to explore it. Kane is thrown into a coma after a disastrous crash landing, but Ross -- encountering the Council and his wife -- realizes they've returned to Earth. Or have they?
1966 books: Tarl Cabot, by John Norman. The world of Gor is the other Earth, where women are slaves.