Moshe ben Maimon (Maimonides) describes the offering process in his Mishneh Torah.1
5 There are three types of individual peace-offerings:
a) a peace-offering that is brought without bread, e.g., the festive peace-offering or the peace-offering of celebration, they are called peace-offerings;
b) peace-offerings brought with bread because of a vow or a pleadge; these are called thanksgiving offerings and the bread is called the bread of the thanksgiving offering;
c) the peace-offering brought by a nazirite on the day he completes his nazirite vow; this offering is accompanied by bread and is called the nazirite's ram.
6 What is procedure for bringing these three [types of offerings]? [The sacrificial animals] should be slaughtered and their blood should be sprinkled on the altar, as we explained. They are skinned and the portions offered on the altar are removed. Afterwards, the meat is cut up and the breast and the right thigh are set aside. The portions to be offered together with the breast and the thigh are placed on the hands of the owners. A priest places his hands below the hands of the owner and performs tenufah with all these items "before God," to the east [of the Altar]. Whenever there is a requirement for tenufah, it is performed to the east [of the Altar].
7 How is tenufah performed? [The items] are taken [to each of the directions] and returned, lifted up and brought low. If the sacrifice was a thanksgiving offering, one should be taken one from each [of the four] groups of ten breads that are brought with it and place it together with the breast, the thigh, and the portions offered on the altar. Tenufah should be performed with all of these items upon the owner's hands, as explained.
8 How are they placed on the owner's hands? The fats are placed on the owner's hands with the breast and the thigh above them. The two kidneys and the lobe of the liver are placed above them and if [the offering includes] bread, it is placed above them and tenufah is performed with all these items.
9 If the sacrifice was a nazirite's ram, [the priest] should remove the portions to be offered on the altar, set aside the breast and the thigh, and cook the remainder of the ram in the Women's Courtyard. The priest takes the cooked foreleg from the ram and one from each [of the two] groups of ten breads that are brought with it, together with the breast, the thigh, and the portions offered on the altar and places everything on the nazirite's hands. The priest places his hands under the owner's hands and moves all [items] as we described.
Alfred Edersheim also wrote,2
Closely connected with this was ‘the lifting and waving’ of certain sacrifices. The priest put his hands under those of the offerer, and moved the sacrifice upwards and downwards, right and left; according to Abarbanel also ‘forwards and backwards.’
1 Mishneh Torah, Sefer Avoda, Hilkhot Maʿaseh ha-Korbanot, Chapter 9, Hilkhot 5–9; also, cf. Babylonian Talmud, Seder Kodashim, Tractate Menachot, Chapter 5, Folio 61a, Mishna: כיצד הוא עושה נותן שתי הלחם על גבי שני כבשים ומניח שתי ידיו למטה מוליך ומביא מעלה ומוריד שנאמר (שמות כט) אשר הונף ואשר הורם
2 Edersheim, p. 88
Edersheim, Alfred. The Temple—Its Ministry and Services, as They Were at the Time of Jesus Christ. London: The Religious Tract Society, 1874.
Moshe ben Maimon (Maimonides). Mishneh Torah. Trans. Touger, Eliyahu. Vol. 22. Brooklyn: Moznaim, 1998.