It is a checklist. Why would the order matter?
Read quickly, this is fifteen jobs with the word "before" between them. The Gemara (the rabbis' discussions built on the Mishnah) reads it slowly. Right after quoting the list, it goes back to the top and asks about step after step: how do we know this one comes first?
Some answers are verses. The Torah says the kohen (priest) burns the incense "when he cleans the lamps." So the lamps are finished first, and then comes the incense.
Other answers are rules for choosing between two good things. The large fire is set up before the small one because it atones for more. The two logs come before the inner altar because what makes another service possible goes first. The lamb's blood comes before the last two lamps because what atones goes first. And once you have started a job, keep going: the kohen who began the woodpile finishes it, and the kohen who began the lamps does most of them.
The best-known rule comes from Reish Lakish, a sage in the Land of Israel: you do not pass by a mitzvah. Why is the ash cleared from the inner altar before the lamps are cleaned? Rava, Abaye's colleague, answers with the floor plan. The table stood on the north side and the menorah on the south. The altar stood between them, a little closer to the entrance, so a kohen walking in reached it first. Rashi (the classic commentator, France, died 1105) explains: whoever meets a mitzvah should not walk past it.
That rule left the Temple. In the same passage Rava applies it to tefillin. Rashi explains: the arm comes before the head, because the arm is what you reach first. The Shulchan Aruch (the main code of Jewish law) applies it to a tallit bag. If your tallit and tefillin share one bag, do not put the tefillin on top. You would meet them first and have to put them on before the tallit.
So the list is a schedule, and also a set of worked examples of what to do first when everything matters.
Yoma 33a-33b, with Rashi,Shulchan Aruch, Orach Chaim 25:1The first line says this follows one rabbi. Is it what really happened?
The opening line is honest about this. Abaye gives the order "following Abba Shaul." Abba Shaul was one of the sages of the Mishnah (the first written collection of the Sages' rulings, about 200 CE), and on this point the other Sages disagreed with him.
The argument is about the middle of the list. The seven lamps of the menorah were cleaned in two rounds, five and then two, with something in between. The Gemara quotes both views. The Sages: he should not clean the lamps and then burn the incense. He burns the incense and then cleans. Abba Shaul: he cleans, and then he burns. In Abaye's list the lamb's blood comes between the two rounds, and the incense waits until all seven lamps are done. For the Sages, the incense is what comes between.
Both sides read the same verse: "every morning, when he cleans the lamps, he shall burn it." Abba Shaul takes the order of the words as the order of the work. The Sages take it as a time: burn the incense at the hour the lamps are cleaned.
The Gemara does not hide the gap. It notices that Abaye's list does not match the way a Mishnah had just been explained, and answers: that one follows Abba Shaul, this one the Sages.
Later the Rambam (Maimonides, died 1204) ruled like the Sages. The Kesef Mishneh, Rabbi Yosef Karo's commentary on the Rambam (1500s), spells out the result: blood, five lamps, incense, two lamps. Another commentary printed there, the Lechem Mishneh, notes that apart from this the Rambam's order is Abaye's.
So why does the siddur keep Abaye's version? Rabbi Yosef Karo asks this himself in his Beit Yosef (his commentary on the Tur, an earlier law code). His answer: people found that Abaye set out the order following Abba Shaul, which implies Abaye held that this is the law. So they did not want to change it.
The sources agree on most of the morning and disagree about a few steps in the middle. The first line tells you which side you are reading.
Yoma 14b-15a,Rambam, Laws of Daily Offerings 6:3-4, with Kesef Mishneh and Lechem Mishneh,Beit Yosef, Orach Chaim 48Abaye never saw the Temple. So how did he know its schedule?
A story card says Abaye never saw the Temple. His first line names his source: "the tradition." Rashi (the classic commentator, France, died 1105) explains: what the whole academy taught. The line traces it no further back. Other pages show how the Temple was remembered.
The Mishnah (the first written collection of the Sages' rulings) has a tractate called Middot, "measurements," that describes the layout of the Temple. The Gemara (the rabbis' discussions built on the Mishnah) names who taught it: Rabbi Eliezer ben Yaakov, a sage of the Mishnah. A guard found asleep on his watch could have his clothes burned, the tractate says. He adds: "One time they found my mother's brother sleeping and burned his clothes."
Twice the tractate reaches a room and quotes him: "I forgot what it was used for." Both times another sage supplies it: a store for wine and oil, and the room of the Kohen Gadol (high priest). That sage is named Abba Shaul, the name in the first line of this list. Reading the two as one man is our step, not something these sources say.
Another tractate, Tamid, walks through the morning service. Rashi points to it for where the second fire in Abaye's list stood. Its walk-through ends: "This is the order of the daily offering in the service of the House of our God. May it be His will that it be rebuilt soon, in our days. Amen."
Why learn Temple laws no one could use? Abaye had an answer. After a ruling on a law of offerings, his colleague Rava objected: "A ruling for the days of the Mashiach (the Messiah)?" Abaye: then no one should teach the laws of offerings. "Rather, study and receive reward." Rava replied that he had only questioned the ruling. Rashi explains: the laws of the service must be learned "because it is Torah."
Abaye says of one step, "The reason I do not know." Rabbi Eliezer ben Yaakov says, "I forgot." Neither fills the gap with a guess. That comparison is ours, and so is setting these sources beside this list.
Yoma 33a, with Rashi,Mishnah Middot 1:2, 2:5 and 5:4,Yoma 16a,Mishnah Tamid 7:3,Zevachim 44b-45a, with Rashi