Halacha requires facing the direction of Eretz Yisrael while reciting the Amida. In the United States, of course, one must face toward the east. It should be noted that this Halacha applies only to a person reciting the Amida. During the Hazzan's repetition, the other congregants listening to the repetition are not required to face toward the east.
If a person realized while reciting the Amida that he is facing the wrong direction, should he turn his body to face the east?
The Halacha in this case depends on which direction the person faces. If he faces to the north or south, then he should continue the Amida in his current position, but turn his head toward the east. If, however, he is facing west, with his back to the east, then he must, in fact, turn around to face the east, even though he is in the middle of the Amida. Similarly, if a person sees his fellow reciting the Amida toward the west, he should draw his attention to this fact and have him turn around to face the proper direction.
The Gemara in Masechet Baba Batra comments, "One who seeks wisdom shall turn to the south; one who seeks wealth shall turn to the north." In the Mishkan and the Bet Ha'mikdash, the Menora, the symbol of Torah wisdom, was positioned along the south wall, while the Shulhan (table), which represents material prosperity, was situated to the north. The Gemara thus advises one who seeks Torah knowledge to turn to the south during prayer in order to draw from the influence of the Menora, whereas one who desires wealth should turn to the north, the side of the Shulhan.
Needless to say, this comment does not conflict with the well-known Halacha requiring a person to face the direction of Eretz Yisrael while he recites the Amida. Rather, it means that one who so wishes can turn slightly to northeast or southeast during the Amida in an effort to obtain either wealth or Torah knowledge, respectively.
Summary: If a person realizes during the Amida that he is facing toward the north or south, rather than to the east, the direction of Eretz Yisrael (in the United States), he should remain in his current position but turn his head to the east. If he realizes he is facing the west, he should turn around to stand in the proper position. The Talmud teaches that one who seeks Torah wisdom should turn slightly to the south during the Amida, and one who seeks wealth should turn slightly to the north.
