How do ashes of the Para Aduma work?

The laws of Para Aduma are notoriously difficult to comprehend? How is it that they render impure a ritually pure individual, yet render pure a ritually impure individual? They should either render everyone equally impure, or render everyone equally pure!

In different times, people may try to explain its functioning in different ways. For example, nowadays, someone might try to explain its functioning as similar to the toggle flip-flop, or functioning like the CAPS LOCK key, where pressing it always changes its state to the opposite.

Ibn Ezra cites the Gaon (Rav Saadia Gaon?) who compares its functioning to that of the impact of honey on people of different humours. Thus:

פרה אדמה תמימה -
שלא תהיה קטנה.

אמר הגאון:

כי מי נדה כמו הדבש שיזיק לבעל המרה האדומה ויועיל לבעל הלחה.

ואין צורך

The Gaon says that the water of sprinkling is like honey, which harms the one of {dominant} red humour {=blood}, and aids the one of {dominant humour of} phlegm. And this is not necessary.

It is not necessary because the Torah explicitly calls it a chok, so we need not find explanations for it.

R' Shmuel Matot in his supercommentary on Ibn Ezra says essentially the same, though continues with his own explanation of the phenomenon {not included to the right}.

"like honey which harms the one of red humour and helps those of phlegmatic humour": and the opinion of the Gaon, z"l, is to give explanation for the purification of the impure and the rendering impure of the pure, in the eyes of the learners, and he wished to bring a comparison from nature, that we find in it things which do both something and its opposite, because of the recipients. And the scholar {=Ibn Ezra} said that "this is not necessary," and his reason is that it is a chok, of the statutes of our Torah.

And perhaps there is to it some pertinence from an aspect of medicine, and one should contemplate this well, since after all it {=Para Aduma} is not a sacrifice. And many say in it things {explanations}, and it is not correct to write them, and I will bring out words from my heart to settle my mind; I write them here. Perhaps they will be considered correct in the eyes of a few of those who hear them.

He goes on to offer an explanation. See inside.

If I {=Josh} might offer my own explanation. First, Midrash Rabba on Chukat:

שאל עובד כוכבים אחד את רבן יוחנן בן זכאי אילין עובדייא דאתון עבדין נראין כמין כשפים
אתם מביאים פרה ושורפין אותה וכותשין אותה ונוטלין את אפרה ואחד מכם מטמא למת מזין עליו ב' וג' טיפין ואתם אומרים לו טהרת
אמר לו לא נכנסה בך רוח תזזית מימיך?
אמר לו לאו
ראית אדם שנכנסה בו רוח תזזית?
אמר לו הן
א"ל ומה אתם עושין לו?
אמר לו מביאין עיקרין ומעשנין תחתיו ומרביצים עליה מים והיא בורחת
א"ל ישמעו אזניך מה שאתה מוצא מפיך
כך הרוח הזו רוח טומאה דכתיב (זכריה יג) וגם את הנביאים ואת רוח הטומאה אעביר מן הארץ מזין עליו מי נדה והוא בורח
לאחר שיצא אמרו לו תלמידיו רבינו לזה דחית בקנה לנו מה אתה אומר?
אמר להם חייכם לא המת מטמא ולא המים מטהרין!
אלא אמר הקב"ה חקה חקקתי גזירה גזרתי אי אתה רשאי לעבור על גזרתי דכתיב זאת חוקת התורה

A certain gentile (idolator) asked Rabban Yochanan ben Zakkai: These acts which you do appear like sorcery!

You bring a heifer and burn it, and crush it, and take its ashes, and if one of you is impure from a corpse you sprinkle upon him two or three drops and you tell him he is pure!

He (Rabban Yochanan ben Zakkai) said to him: Have you ever had a ruach tezazit {spirit of delirious fever} enter you in your life?

He said to him: No

He (Rabban Yochanan ben Zakkai) said to him: Have you ever seen someone who a ruach tezazit had entered him?

He said to him: Yes.

He (Rabban Yochanan ben Zakkai) said to him: And what do you do for him?

He said to him: We bring {medicinal} roots and cause them to smoke under him, and we inundate it with water, and it {the ruach tezazit} flees.

After he {the idolator} left, his students said to him (Rabban Yochanan ben Zakkai): To this one {the idolator} you have pushed away with a reed {that is, engaged in apologetics}; to us {who know better}, what will you say?

He said to them: By your lives! The corpse does not cause tum`a and the water {of the ashes of the Parah Adumah} do not cause him to be tahor!

Rather, the Holy One, Blessed Be He, said: A chok I have established, a decree I have decreed, and you are not permitted to violate my decree. As it states {in the second verse of parashat Chukat}: זֹאת חֻקַּת הַתּוֹרָה - "This is the statute of the law."

Which gets it from Midrash Tanchuma on Chukas. So if tumah and tahara are not real, but rather artificial constructs from On High, then why is Hashem imposing this chok? Perhaps just to be arbitrary. But I will violate the ethos of this and propose a reason anyway.

Each kohen is not being rendered impure in the sense of spiritually affected by this. However, a tamei kohen cannot perform any avodah, including the next avodah in the series of steps for making the Parah Aduma. This is then a mechanism which adds specialness to the process. Each step is performed by a different kohen, and then he is down for the count. Only until evening, enough to prevent him from performing another step. And this makes it more solemn, and less a do-it-yourself affair.