Rashi in Parshat Chukat quotes a Midrash that explains that the red heifer is a tikun for the sin of the golden calf.   "To what can this be likened?  To a maid that worked in the palace of the king.   One day her child came and soiled the palace with his filth.  The king said:  'Let the mother come and clean up her child's filth' "
This is the concept of atonement for the golden calf, for the Holy One said: "Let the red heifer come and atone for the golden calf."

The Midrash examines the concept of the red heifer and the golden calf, and finds many striking parallels and connections.  For example:  

• Why must the heifer be red?  Scripture likens sin to red, because when a person sins, he forfeits his blood: "Though your sins be as scarlet, they shall be as white as snow;  though they be like red crimson, they shall be white as wool."   (Isaiah 1:18)   Says the Holy One:  Since Israel's sins are red, let the heifer also be red - and when it is burned, it's ashes are white ... thus the prophet assures his people in the Creator's name that God promises "they shall be white as snow."
• In order for the heifer to be fit,  it must never have carried a yoke.  This is an allusion to Israel who threw off the yoke of Heaven in insubordination when they worshiped the golden calf.
• Why must the heifer be given to Eleazar, an assistant, and not the High Priest himself?    Because Aaron oversaw the creation of the golden calf, and therefore it would not seem proper for him to officiate with the heifer.   There is a principle:  "the same one who was prosecutor, cannot become defense attorney."
• The heifer is burned, an allusion to the calf that was burned:  "And he took the calf which they had made, and burned it in the fire...."  Ex 32:20
• Three species are used: hyssop, cedar, and scarlet wool, which are reminiscent of the 3,000 who fell at the sin of the calf.  And why these three?   The cedar is the highest, and the hyssop is the lowest ... and whoever is haughty (equal to the sin of idolatry) must become like a worm (from which the red dye is obtained) in his own eyes, as King David prayed: "I am a worm and not a man"  (Psalm 22:7)   If he will lower and humble himself, his sins will be atoned.
• Just as the sin of the golden calf exerts it's influence forever, as it states: "on that day I will remember you and I will remember your sin"   (Ex 32:34), so too the Holy One commanded that the ashes of the red heifer be kept as a remembrance for all generations:  "And they shall be for the congregation of the children of Israel  for a remembrance." And just as the golden calf rendered all those who participated in it impure, - for idolatry causes impurity, as it is written:  "You shall cast it away as a thing impure ....  "  (Isaiah 30:22), so the heifer renders all those who come into contact with it impure.    And as Israel became pure when Moses burned the golden calf - fire being the symbol of purging sin - so Israel becomes pure through the the burning of the red heifer.

Thus, we have received an explanation that the purification by the heifers's ashes is related to the golden calf episode, and "let the mother come and clean here child's filth", then why is the ordinance of the red heifer still called a "chok"?   Surely the element of mystery has abated; it is simply a matter of rectifying the sin of the golden calf!   Why did King Solomon state regarding the red heifer:  "I said I will get wisdom, but it is beyond me."  (Ecclesiastes)  

Yes, even if it is granted that the secret mechanism which powers the cleansing of these ashes is the power of teshuva, but this itself  is above the realm of comprehension!   For even the greatest wisdom and comprehension cannot fathom or explain the power of teshuva and how it works ...  only the Holy One Himself, in his infinite mercy decreed that is should be so.  Teshuva is the most mysterious Godly secret of all!