Why did Aharon now enter the Kodesh Kodshim for the second time?
Rashi: To take out the Kaf u'Machtah (the bowl and the pan) 1 which he had brought in earlier with the Ketores.
Ramban: It is otherwise inconceivable that he should enter the Heichal merely to remove his clothes.
Rashi writes that the verses are out of order. What forces us to say so?
Rashi, Ramban (based on Yoma, 32b): We have a tradition that on Yom Kipur, the Kohen Gadol immerses five times and washes his hands and feet 10 times 1 ! Rather, we must invert the Pesukim. We insert the Olah of Aharon and that of the people (in Pasuk 24) here, so that Aharon immerses, changes to Bigdei Zahav and brings the Olos 2 and afterwards changes back, before he finally entered to take out the Kaf and Machtah.
Kol Eliyahu, Divrei Eliyahu: Until now, the Torah discussed Aharon's Heter to enter the Kodesh ha'Kodoshim even not on Yom Kipur, with the Korbanos mentioned here (Vayikra Rabah 21:7). Then, he need not immerse five times; everything is in order. After Aharon [died], it is permitted only on Yom Kipur, 3 and the Kohen Gadol must immerse five times, so we must change the order to obligate another change of garments.
Riva: If all Avodas Panim was at once, he would need to wear Bigdei Lavan only once, and need only three Tevilos! (Now, he must wear them twice, and the Bigdei Zahav three times. He immerses before wearing Bigdei Kehunah each time, and washes hands and feet each time he wears them and each time he removes them. - PF)
And some of the Musafim (Rashi).
Kol Eliyahu, Divrei Eliyahu: This is why it mentions 'on Yom Kipur' and 'once a year only now.
Why does the Pasuk invert the order of events?
Ramban: Because it is currently discusses Avodos ha'Yom in the Bigdei Lavan, 1 which it completes (before discussing Avodah in the Bigdei Zahav, e.g. the rams for Olos).
Refer to 16:23:2:2.
Which Avodos were done after the last Tevilah?
Rashi (from Toras Kohanim): Aharon brought some of the Musafim before he took out the Kaf and the Machtah. After taking them out, he immersed for the last time and offered the rest of the Musafim, the afternoon Tamid and afternoon Ketores wearing the Bigdei Zahav. 1
Aharon brought the Olos after his Chatas, since a Chatas always precedes an Olah, but why did he bring them after the Sa'ir la'Azazel?
What are the ramifications of "Vehinicham Sham", mentioned in connection with the four white Begadim that Aharon wore on Yom Kipur?
Rashi and Seforno: It means that, after removing the Kaf and Machtah, he was to change into the eight Begadim of a Kohen Gadol, to leave the white ones there 1 , never to be used again, 2 not by a Kohen Hedyot during the year nor by Aharon on the following Yom Kipur.
Rashi writes that the verses are out of order. Within one matter, the Torah is in order!
Moshav Zekenim (from Yuma 32a): Even within one matter, when there is a proof, the Torah is not in order.
In Yuma 32a, it asks why Aharon entered, and says 'to take out the spoon and pan.' The verse says that he entered the Ohel Mo'ed to remove his garments! And the spoon and pan were in the Kodesh Kodashim!
Ramban: We cannot say that he enters merely to remove his clothes! Rather, it was for Avodah, i.e. to take out the bowl and the pan, which he had brought in earlier with the Ketores.
Rashi writes that he enters afterwards to take out the Kaf and the Machtah. One may not enter without need! He could leave them there, and next year he will enter with a new pan, and when he leaves, he will take out the old pan!
Riva citing R. Tam of Orai': It would be disgraceful to leave the empty pan there. Chachamim permitted moving a pan with ashes for it is repulsive to people. All the more so, we should remove it from in front of Hashem!