Gemara

Question: Is Holachah (of blood) with the left hand Kosher?

Answer (Rav Sheshes - Mishnah): The Kohen Gadol carries the incense pan in his right hand and the spoon (with Ketores) in his left hand. (This is like Holachah!)

Question: Why didn't Rav Sheshes learn from the Mishnah of taking limbs to the Mizbe'ach?

(Mishnah): One Kohen holds (the head in his right hand and) the right hind leg in his left hand.

Answer: One might have thought that Holachah which is not Me'akev Kaparah is Kosher in the left hand, but not Holachah which is Me'akev Kaparah. Therefore, he learns from the Ketores of Yom Kipur which is Me'akev Kaparah.

Question (Beraisa): Kabalah, Holachah and Zerikah are Pasul if done by a Zar, Onen, someone drunk, or a Ba'al Mum. Also, if the Kohen sat or used his left hand they are Pasul.

Rav Sheshes is refuted.

Zevachim 32a - Mishnah: If Kabalah was done in the right hand and the Keli was transferred to the left hand, he returns it to the right hand. (Similarly, if it was given to a Pasul or put into a Chulin Keli, it is returned to a Kosher Kohen in a Kli Shares.)

34b: The Mishnah needed to teach all three cases:

Had it taught only giving it to a Pasul, one might have thought that this refers only to a Tamei, who is Kosher for Avodas Tzibur (if no Tahor Kohen is available, but if it was transferred to the left hand, the Korban is Pasul);

Had it taught only transferring to the left hand, one might have thought that this can be fixed because the left hand is used for Avodah on Yom Kipur (but if it was put into a Chulin Keli, the Korban is Pasul).

Menachos 25a: The Tzitz bears the sin of Tum'ah, for Tum'ah is wholly permitted to the Tzibur.

Question (R. Ila'a): Perhaps it bears the sin of Avodah done with the left hand, for this is wholly permitted on Yom Kipur (regarding carrying the Ketores)!

Answer (Abaye): It says "Avon", i.e. something that is always forbidden, but is Nidcheh (permitted b'Di'eved) to the Tzibur. The left hand is used l'Chatchilah on Yom Kipur.

Rishonim

Rambam (Hilchos Bi'as Makdish 5:18): Avodah is only with the right hand. If one did Avodah with the left hand it is Pasul, but he is not lashed. If Kabalah was done with the right hand and the left hand helped it is Kosher, for help is insignificant.

Rambam (Hilchos Avodas Yom Kipur 4:1): We already explained that Holachah with the left hand is Pasul regarding blood and other Avodos. Letter of the law, the Kohen Gadol should carry the pan in his left hand and the spoon with Ketores in his right hand. However, he cannot carry the pan in his left hand until the Aron because it is too heavy and hot. Therefore he carries the pan in his right hand and the spoon in his left hand.

Question: If Avodah with the left hand is Pasul, why is it permitted just because it is too hard the other way?

Answer #1 (Ri Korkus): The Torah requires the Kohen Gadol to take both together ("V'Lokach Melo ha'Machtah...u'Melo Chafnav...v'Hevi"). Not every Kohen can carry the pan in his left hand, because it is heavy. We must say that the Torah allows carrying the Ketores in the left hand. We need not seek a Kohen able to carry the pan in his left hand. It is always possible to carry blood in the right hand. Tosfos (Yoma 48b DH Holachah and Menachos 25a DH v'Eima Avon Smol) hold like this answer.

Answer #2 (Ri Korkus): Carrying the Ketores is not a primary Avodah. It is like carrying limbs to the Mizbe'ach, which is Kosher with the left hand for this reason. However, on 47a it appears that carrying the Ketores is a primary Avodah (the Gemara was astounded why it is done with the left hand).

Answer #3 (Kesef Mishneh): Holachah with the left hand is Pasul only mid'Rabanan.

Question: The Gemara in Menachos (25a) suggests that the Tzitz bears the sin of Avodah done with the left hand, for this is wholly permitted on Yom Kipur. This shows that it is normally forbidden mid'Oraisa!

Answer (Tal Torah 48b): Perhaps the Gemara refers to Chafinah (taking a double handful of Ketores).

Rebuttal (of Kesef Mishneh - Mayim Chayim, brought in Likutim of Frankel Rambam Avodas Yom Kipur 4:1 DH Hayah): If Holachah with the left hand is Pasul, this is because the laws of Kabalah apply to it (Zevachim 4a). This is a Pesul mid'Oraisa!

Answer #4 (Kesef Mishneh): Regarding Holachas Dam, the left hand is Pasul because Holachah is essential. Nevertheless the left hand is not Pasul for Holachah of the Ketores.

Lechem Mishneh: Rav Sheshes disagreed with these answers, therefore he concluded that the left hand is Kosher for Holachah.

Answer #5 (Ritva 49a DH Teyuvta): (Even though the left hand is normally Pasul,) it is permitted for carrying the Ketores because a Keli is not required, just it is impossible otherwise.

Tal Torah (ibid.): This means, since the Torah allowed the left hand (together with the right hand) for Chafinah, and letter of the law it should be permitted to bring the Ketores without a Keli in his hands, this shows that the Torah allowed to use the left hand for this. (A Keli must be used, for otherwise no hand would be free to carry the Machtah.)

Tosfos (Zevachim 25a DH v'Harei (last skinny lines)): Surely, Holachah with the left hand is Kosher, for it does not say 'Etzba' (elsewhere 'Etzba' teaches that Yamin is Me'akev). Perhaps he must return the blood to the right hand lest he do Zerikah with the left hand. Alternatively, l'Chatchilah Holachah must be with the right hand.

Acharonim

Tosfos Yom Tov (Yoma 5:1 Sof DH Notal): Carrying the Machtah is only Machshirei Avodah. There is no reason why it should require the right hand.

Or Some'ach (Avodas Yom Kipur 4:1 DH Divrei): The Torah did not mention the spoon in which the Ketores is carried to teach that this is not an important Avodah, hence the left hand is Kosher. Similarly, the Torah did not mention dipping the finger in blood of an outer Chatas, to teach that this is not an important Avodah, hence it cannot Mefagel.