Every year at this time we seem to run into the same questions about the significance of the mandatum at the Mass of the Lord's Supper (or the washing of the feet). (Mandatum means command signifying the Lord's command to his disciples to serve others in the same way he has served them; it is also where we get the name Maunday Thursday.) More to the point is the question about whose feet should be washed. As with any matter liturgical, the answer must begin with a proper understanding of the theological significance of the event. Many have written on this, and I will not attempt here a complete answer. Instead, I offer a summary along with the rubric from the Roman Missal and a few links that offers a more complete discussion of the problem.
The significance of the event is a re-presentation of Christ's washing of the disciples feet the night before he died. Recall that the Last Supper also marks the institution of the Eucharist and the holy priesthood. Christ, who came to serve and not to be served, tells the disciples, as priests, to do unto others as he has done unto them. The ritual of the feet washing has often been mischaracterized as an act of service and charity. While service and charity are certainly a part of the call to priesthood, to reduce the significance of the feet washing to merely an act of kindness is to lose the entire point of the ritual. The ritual performed by Christ has more to do with a priestly ordination rather than priestly service. The act was the fulfillment of an Old Testament ordination ritual. The ritual is described in Leviticus 8:6-7, when Aaron and his sons were consecrated as priests for the Temple:
"And Moses brought Aaron and his sons, and washed them with water. And he put upon him the coat, and girded him with the girdle, and clothed him with the robe, and put the ephod upon him, and he girded him with the curious girdle of the ephod."
We also read in the Book of Exodus (30:18-20):
"The LORD said to Moses, 'For ablutions you shall make a bronze laver with a bronze base. Place it between the meeting tent and the altar, and put water in it. Aaron and his sons shall use it in washing their hands and feet. When they are about to enter the meeting tent, they must wash with water, lest they die."
In John 13:5, Jesus "poured water into a basin and began to wash the disciples' feet and dry them with the towel around his waist," a direct parallel to the old testament ritual.