After the Levites were cleansed, they were presented to God. Verses 9 through 22 describe this presenting of the Levites.
"You shall bring the Levites before the tent of meeting, and gather the whole assembly of the sons of Israel. And you shall bring the Levites before Jehovah, and the sons of Israel shall lay their hands upon the Levites" (vv. 9-10). Here we see that the sons of Israel laid their hands upon the Levites. This means that the sons of Israel identified themselves with the Levites.
Actually, the services of the tabernacle should have been taken care of by the sons of Israel, yet God chose the Levites to replace the sons of Israel. Now these Levites were about to be presented to God, meaning that all the sons of Israel would be presented to God. Hence, the sons of Israel laid their hands upon the Levites to identify themselves with the Levites.
"Aaron shall offer the Levites before Jehovah as a wave offering from the sons of Israel, that they may do the service of Jehovah" (v. 11). In typology, a wave offering signifies the resurrected Christ. Here the offering was not composed of cattle but of living persons, the Levites. The Levites were offered as a wave offering that they might do the service of God.
"Then the Levites shall lay their hands upon the heads of the bulls; and you shall offer the one for a sin offering and the other for a burnt offering to Jehovah, to make propitiation for the Levites" (v. 12). For the Levites to lay their hands on the bulls meant that they identified themselves with the bulls. Here three parties were identified with one another through the laying on of hands: the sons of Israel, the Levites, and the bulls.
Moses offered one bull for a sin offering and the other for a burnt offering to make propitiation for the Levites. The Levites had been cleansed, but they were still held condemned because they still had sin. On the one hand, Aaron could offer them to God, yet, on the other hand, Moses had to make propitiation for them.