Gemara

Question: Zerikas Olah is done on the two opposite corners (that have a Yesod). Why do we begin with the northeast and not with the southwest?

Answer: We learned that one always turns to the right. (For Zerikas Chatas one ascends the ramp in the south, and turns to the right, i.e. east. We likewise begin Zerikas Olah in the east.)

Mishnah 43b: Every day Kohanim (who have Avodah to do on top of the Mizbe'ach) ascend the ramp in the east and descend in the west.

45a: This is because one always turns to the right (they circle the Mizbe'ach counterclockwise; we minimize the amount they must walk on top of the Mizbe'ach).

Zevachim 63a: There are three exceptions, i.e. Avodos done in the southwest corner. For these, one ascends on the left and descends the same way.

Sotah 15b - Mishnah: The Kohen enters the Heichal and turns to his right (to take dirt for Mei Sotah).

This is because one always turns to the right.

Zevachim 62b - Beraisa: The Yam Shel Shlomo rested on 12 supports - "Sheloshah Ponim Tzafonah u'Sheloshah Ponim Yamah... Ponim Negbah... Ponim Mizrachah" (the verse lists them in the order of one who starts in the north and circles around to his right);

This teaches that whenever we circle around, it is to the right.

Question: The verse teaches about the supports, we cannot learn anything else from it!

Answer: It was not necessary to repeat Ponim each time. The repetition teaches that we always circle to the right.

Midos 2:2: Anyone who enters Har ha'Bayis turns to his right and goes around (counter-clockwise). The only exceptions are a mourner or someone excommunicated. He turns to his left and goes around (clockwise), anyone who sees him asks what happened, and blesses him accordingly.

Rishonim

Question: Why do we learn from the Yam Shel Shlomo? Perhaps we should rather learn from the Migrash around the Leviyim's cities - the Torah mentions the directions east, south...i.e. circling to the left (for one who is facing the city)!

Answer (Tosfos 17b DH v'Ha): "Ponim" is extra to teach that we learn from the Yam Shel Shlomo.

Rambam (Hilchos Nesi'as Kapayim 14:13): When Kohanim turn to face the Tzibur and when they turn away from the Tzibur they go to their right (clockwise). This is because one always turns to the right.

Poskim

Rema (OC 95:1): Some say that when one stands up for Tefilah he takes three steps forward, like one who approaches a matter he needs to do.

Beis Yosef (OC 111 DH v'Chasuv citing Orchos Chayim): The three steps before Tefilah begin with the right foot. This is because one always turns to the right (Yoma 58b), and one moves his right foot first when he begins to walk (Menachos 34a).

Shulchan Aruch (OC 128:17): When Kohanim turn around before and after Birkas Kohanim they go to their right.

Shulchan Aruch (141:7): One called for an Aliyah l'Torah takes the shortest path to the Bimah and returns the longer way. If both paths are equal he takes the right path and returns the opposite way.

Source (Beis Yosef DH Kasuv b'Kisvei): One enters the Azarah in the short path and leaves the long way. The Terumas ha'Deshen (2:119) learns from here to going to the Bimah. If the paths are equal, we adopt the rule of always turning to the right. Megilah (29a) teaches that one should leave a Beis ha'Keneses in a different direction than he entered.

Rebuttal (Gra DH Kol): There is no Tosefta or Gemara saying that we enter the Azarah in the short path and leave the long way! Even if there was, it would be better to learn from the Mizbe'ach; we leave in the shortest way (Zevachim 63).

Note: Presumably, the Terumas ha'Deshen learned from Megilah 29a, which learns (about leaving a Beis ha'Keneses in a different direction than he entered) from the Azarah. The Gra himself says there that the text should say that it is a Mitzvah, not merely permitted! The Gra's objection is that taking a short or long path is not the same as leaving in a different direction.

Mishnah Berurah (22,23,25): One goes promptly in the shortest path because of Kavod ha'Tzibur (they should not wait for him) and to show eagerness for the Mitzvah. He leaves on the long path so it will not look like the Mitzvah was a burden for him.

Beis Yosef (128 DH uchshe'Machazirin): Likewise, when the Shali'ach Tzibur takes the Torah from the Aron in the east (and faces the Bimah in the west), he should turn right and go to the Bimah via the north. To return the Torah he turns right and goes via the south.

Magen Avraham (141:7): The Shali'ach Tzibur should take the shortest path.

Kaf ha'Chayim (37): Someone called to the Torah takes the shortest path because of Kavod ha'Tzibur or to show eagerness for the Mitzvah. These reasons do not apply to the Shali'ach Tzibur.

Shulchan Aruch (551:10): We wave the Lulav east, south, west, north.

Objection (Levush, brought in Taz 13): We should go the other way, like Kohanim on top of the Mizbe'ach!

Answer (Taz 13): There also, the Kohen begins to his right (but after reaching the northeast corner he is forced to turn left)!

Magen Avraham (21): A left-handed person does like everyone else. We learn from Midos 2:2. We assume that anyone going to the left is an Avel or in Niduy, we do not say that he is left-handed!

Kaf ha'Chayim (49,103): In Yerushalayim the custom is to wave south, north, east, up, down, and west like the Ari Zal did, based on the Sefiros associated with these directions.

Shulchan Aruch (561:1): Hakafos (with the Lulav) are done to the right.

Magen Avraham (1): Everyone faces the Torah in the middle on the Bimah, therefore we go east-north... (counterclockwise).

Shulchan Aruch (676:5): On the first night of Chanukah one lights the Ner on the right. Each successive night he adds a Ner to the left and blesses on (begins with) it, in order to turn to the right (to light the others). This shows that there was an additional Nes each additional day.

Terumas ha'Deshen (106): Benei Austrich light from right to left. Perhaps they consider this like turning to the right. Alternatively, because they have no Mezuzah in the room they must begin on the right.

Rebuttal (Beis Yosef DH uv'Terumas): In any case one should begin on the left.

Taz (6): 'Turning to the right' means to begin with the right. There is no reason to favor the latter Neros over the first! Also, if one begins on the left he passes over Mitzvos (the Neros that are closer to his right hand)!

Mishnah Berurah (11): One should stand to the left to avoid passing over Mitzvos.

Gra (DH va'Yevarech): The primary Mitzvah is the Ner closest to the doorpost, the others are merely for Mehadrin Min ha'Mehadrin. One should first do the primary Mitzvah, whether it is on the right or left.

Bi'ur Halachah (DH uva'Leil): The Shulchan Aruch holds that turning to the right overrides all other considerations.

Mishnah Berurah (9): Whichever way one starts is acceptable. It is best to arrange the Neros along the thickness of the doorway, so all are within a Tefach of the doorpost (Maharshal).

Chasam Sofer (Teshuvos OC 187, DH ha'Oleh): Turning to the right overrides all other considerations. However, since the Beis Yosef disagrees one should position himself from the beginning so he will not need to pass over Mitzvos or take a longer path.