Well, I can't say much about the symmetry you refer to. Tolkien played around with the numbers of the Great Rings while writing LOTR and he settled on 1, 3, 7, 9. However, in constructing the history of the Rings he established firmly several points:

1) The Elves made many "lesser" Rings before attempting the Great Rings.

2) All of the Rings of Power, except for the One (and Saruman's, if it was indeed a ROP he made himself), were made BY the Elves FOR the Elves.

3) The Seven and the Nine were given out after Sauron captured them.

What is the significance of the Seven and Nine? We are not told. The Nine were in Ost-en-Edhil when Sauron took the city, according to a rejected history of Galadriel and Celeborn. We have no other account of the fall of Ost-en-Edhil to help us determine whether Tolkien intended to retain that point. But it may be that the Seven and Nine ultimately became "Seven" and "Nine" because of how they were distributed, if Sauron first gave Rings of Power to the Dwarves and chose to give only one to each of the Dwarf kings.

There is much we don't know about the Rings, however.

------------------
Parma Endorion: Essays on Middle-earth, Revised Edition