As we saw last night, Holy Thursday is about the institution of the Eucharist and the New Covenant; but right before then, it’s about the institution of the priesthood, and the obligation for the new priestly people, Christians, to act as servants toward each other.
Jesus takes off His himation, or cloak, which leaves Him in His long seamless robe; and then He ties a towel around Himself as a belt.
And what did the Jewish priests wear at the Temple? A long seamless linen robe down to the ankles, and a belt or sash tied around the robe.
But the way Jesus was doing it (ie, with the towel) seems to have been how servants did the footwashing for guests. So it was a sort of visual pun.
Leviticus 8:6 – “Moses had Aaron and his sons come forward, and he washed them. He put the linen robe on Aaron and fastened the belt around him.” (This is when Aaron and his sons are made priests of the Lord.)
Exodus 40:30-32 – “He placed the basin between the Tent of Meeting and the altar, and put water in it for washing; and from it, Moses, Aaron, and his sons washed their hands and their feet. They washed whenever they entered the Tent of Meeting or approached the altar, just as the LORD had commanded Moses.
You get another priestly thing when Jesus warns Peter that unless he is washed, “You do not have a portion with Me.” (John 13:8) The Levites and priests of Israel were the only tribe that did not have a portion of the land; their portion and inheritance was God Himself.
(Oh, and here’s something funny. Peter didn’t just say, “Lord, you will _never_ wash my feet.” He said, “You will not wash my feet eis ton aiona,” which literally means, “to the age,” and is a translation of some kind of Hebrew expression about “olam.” It’s something like “ever in my life,” “in this age,” or even “forever.” So it’s a pretty exaggerated or emotional way to talk. Jesus says it because He’s God and He can, but Peter saying it is silly.)