ייִדישער מחשבֿה אויף במדבר 19:9

The red heifer is called a sin-offering, because it effects the purification of persons that have become unclean through the dead body of a human being, and enables them to enter the Sanctuary [and to eat of hallowed things]. The idea taught by this law is this: Those who have defiled themselves would never be allowed to enter the Sanctuary, or to partake of holy things, were it not for the sacrifice of the red heifer, by which this sin is removed; in the same manner as the plate [which the high-priest wears on his forehead] atones for uncleanness, and as a similar object is attained by the goats that are burnt. For this reason those were unclean who were engaged in the sacrifice of the heifer or the goats which were burnt, and even their garments were unclean. The same was the law in the case of the goat that was sent away [on the Day of Atonement]; for it was believed that it made unclean those who touched it, because it carried off so many sins.