R' Samson Raphael Hirsch, in his grand exposition of the symbolism in the red heifer procedure1, relates the third and seventh days of the procedure to the Third and Seventh Days of Creation and to the basic regulations of Man that were introduced on each.
The point of the procedure, according to R' Hirsch, is to restore the patient's sense of human free-willed ability to choose service of God over sin. This sense had been damaged by the patient's contact with a dead body, which evokes a strong sense of humans as strictly physical beings, strictly compelled by the laws of nature. The free-willed submission to God's will that the procedure revives applies to two basic aspects of human existence: physical and mental/spiritual.
The very basest physical activities that humans engage in - consumption, growth, reproduction, etc., we share with all living things, starting (as far as the Torah's categories in the Creation account are concerned) with plant life. In the Creation account, God Created plants on the Third Day and immediately applied a law to regulate their reproduction - "לְמִינוֹ" - "for its own species" (Genesis 1:11). In his commentary on Genesis 1:11-13, R' Hirsch, channeling the Talmud in Chulin 60a, says that this law of heredity, which the plants follow as a matter of natural law, is meant to be an example to humans of all the checks on our own physical activities that laws of morality are meant to impose, should we choose to submit to them. So we sprinkle on the third day of the procedure to restore our sense of free will in submitting to God's restrictions on our physical activities.
On the Seventh Day, God stopped His own Creation activity, thus setting an example for humans' duty to regulate the aspect of ourselves that'd Godly - our ability to impose our will on the world around us through creative acts.2 So, we sprinkle again on the seventh day of the procdeure to restore our sense of free will in submitting to God's restrictions on our high-level, mental, spiritual, willful activities.
The sprinking must occur in these two stages in order, since the human has to first assume free-willed control over his/her physical powers before assuming control over his/her mental/spiritual powers.
1. Printed in his commentary on the Torah following Numbers 19:22 and summarized in this previous Mi Yodeya answer. As always, see the commentary itself for much more detail and beautiful writing.
2. See R' Hirsch's commentary on Genesis 2:3.