Menachos Chapter 1, Mishnah 5. The Mishnah continues to discuss the practical details of preparing a minchah, and it presents two further procedures that can apply to it: hagashah and tenufah.
Hagashah:
Literally, bringing near. One takes the sacred vessel, the vessel in which the ingredients of the minchah are placed before the kemitzah, and brings it to the Altar - touching it to the southwest corner of the Altar. Only afterward is the kemitzah performed. This hagashah applies to most menachos, and it is unique to them: no animal offerings require hagashah, only menachos.
Tenufah:
This means waving, quite literally: one moves the offering back and forth in six directions - the four directions of the compass (forward, backward and side to side), as well as up and down, just as is done when taking the lulav on Sukkos. Most menachos do not require tenufah, but there are a few that do, and there are also other items besides menachos that require tenufah.
The Mishnah sets up the situation as it does in many places: since we have two variables, hagashah and tenufah, there are four possibilities:
Menachos that require hagashah but do not require tenufah.
Menachos that require tenufah but do not require hagashah.
Menachos that require both hagashah and tenufah.
Menachos that require neither hagashah nor tenufah.
Our Mishnah will discuss the most common case, the one that applies to most menachos: those that require hagashah before the kemitzah and do not require tenufah. The next Mishnah will discuss the remaining possibilities.
The words of the Mishnah: "Yesh te'unos hagashah ve'einan te'unos tenufah" - certain menachos require hagashah but do not require waving. "Tenufah velo hagashah" - there are those that require waving but do not require being brought to the southwest corner of the Altar. "Hagashah utenufah" - there are those that require both. "Velo tenufah velo hagashah" - and there are those that require neither one nor the other.
"Ve'elu te'unos hagashah ve'einan te'unos tenufah":
Here is the list of the menachos that require hagashah - which is the standard rule - and do not require tenufah, which is likewise the standard rule. The list mirrors the very same list that appeared several mishnayos earlier, and it applies to most menachos:
Minchas ha'soles - the voluntary meal offering that is neither fried nor baked; it consists only of flour with the addition of olive oil and frankincense. It requires hagashah before the kometz is taken, and no tenufah.
The machavas and the marcheshes - the two fried voluntary meal offerings: the machavas in a flat pan, and the marcheshes in a deep vessel. They too require hagashah before the kemitzah, and they have no tenufah.
Challos and rekikin - the baked meal offerings brought as a donation. The challos are a thicker matzah (something like Sephardic matzah), and the rekikin are thin like crackers, similar to Ashkenazic matzah.
Minchas kohanim - as mentioned earlier, this is not a distinct offering of its own, but rather a kohen who brings one of the five types of menachos that were listed. When a kohen brings it, there are no leftovers eaten by the kohanim; the entire offering is burned on the Altar. Although one might have thought that its law would differ because of the law of the leftovers, its law is the same in this regard: it requires hagashah and does not require tenufah.
Minchas kohen mashiach - this is the minchas chavitin that the Kohen Gadol brings every day, half in the morning and half in the afternoon. It too requires hagashah and has no tenufah.
Minchas goyim - a meal offering brought by a non-Jew. This ruling too follows the view of Rabbi Yose HaGelili, who holds that non-Jews may bring menachos, and in his opinion they require hagashah and not tenufah. However, the halachah follows Rabbi Akiva, that non-Jews do not bring menachos - they bring burnt offerings but not meal offerings - and therefore this ruling is not practiced in halachah.
Minchas nashim - a woman who brings one of the five voluntary meal offerings. Although one might have thought that the law of her minchah would be different, that is not the case: its law is entirely the same, and it requires hagashah and not tenufah.
Minchas chotei - one who brings this offering on account of one of the transgressions listed in Vayikra 5, whose offering is a korban oleh v'yored. One whose means do not allow him to bring an animal sin offering, and not even two birds, brings this offering of plain fine flour with no oil and no frankincense at all, and it requires hagashah.
The View of Rabbi Shimon:
Rabbi Shimon disagrees and holds: "Minchas kohen u'minchas kohen hamashiach ein bahen hagashah, lefi she'ein bahen kemitzah, v'chol she'ein bahen kemitzah ein bahen hagashah" - the meal offering of a kohen and the meal offering of the Anointed Kohen do not require hagashah (bringing the offering near to the corner of the Altar), since they do not have kemitzah (removal of a fistful), and whatever does not have kemitzah does not have hagashah. These two meal offerings do not require hagashah, because they do not have kemitzah.
The basis of his view: kemitzah is performed almost always in order to permit the remainder for eating, and Rabbi Shimon holds that hagashah is bound up with this closely. When there is no fistful coming to permit consumption by people, there is no need for the hagashah that precedes it. The meal offering of a kohen and the meal offering of the Anointed Kohen are burned entirely and the kohen does not eat from their remainder, and therefore they have no kemitzah, and consequently no hagashah either.
The halachah, however, does not follow his opinion: hagashah is required even where there is no kemitzah.