In the end of the Amidah, in Ose Shalom, we bow to one side, to the other side and to the front, this is also done in the end of Lecha Dodi. I've always done this starting with left and then right (in both cases). But I've seen that some people do it the other way around.
Is there a correct order for bowing in these two cases?
For Lecha Dodi yeshiva.org.il says:
Regarding bowing, R. Yakov Emdin brings that his father, the Chacham Zvi, would bow three times when saying Bo’ee Kalah, to his left, then his right and then in front. This is identical to the bowings mentioned when we finish Shmoneh Esreh. The idea is because the Shchina (Divine Presence) is, as if it were, before us (“I have set Hashem always before me”, Tehilim 16, 8), so we bow to Her right (… from His Right-hand (came the) fiery law”, Dvarim 33, 2) before Her left, and then, as if it were, to Her.
A significant difference should be pointed out, that the bowings at the end of Shmoneh Esreh are cited in the Talmud (Yoma 53b) and the Shulchan Aruch (Or. Ch. 123, 1) and are universally accepted, as opposed to those at the end of Lecha Dodi, which is not cited by most of the poskim and is a relatively recent custom of only some communities.
Shulchan Aruch OC 123:1 says first left, then right.
כּוֹרֵעַ וּפוֹסֵעַ ג' פְּסִיעוֹת לְאַחֲרָיו, בִּכְרִיעָה אַחַת, וְאַחַר שֶׁפָּסַע ג' פְּסִיעוֹת בְּעוֹדוֹ כּוֹרֵעַ, קֹדֶם שֶׁיִּזְקֹף, כְּשֶׁיֹּאמַר עוֹשֶׂה שָׁלוֹם בִּמְרוֹמָיו, הוֹפֵךְ פָּנָיו לְצַד שְׂמֹאלוֹ; וּכְשֶׁיֹּאמַר הוּא יַעֲשֶׂה שָׁלוֹם עָלֵינוּ, הוֹפֵךְ פָּנָיו לְצַד יְמִינוֹ; וְאַחַר כָּךְ יִשְׁתַּחֲוֶה לְפָנָיו, כְּעֶבֶד הַנִּפְטָר מֵרַבּוֹ.
One bows and takes three steps backwards, all while still bowing. After the three steps, while still bowing and before straightening up: while saying "Oseh Shalom B'mromav" - turn one's head to the left; "Ya'aseh Shalom" - turn one's head to the right; then bow forward like a servant taking leave of the master.
