These days I am studying mirror symmetry. I am reading "String theory on Calabi-Yau manifolds (hep-th/9702155)" by Brian Greene and Mirror Symmetry from Clay Mathematics Institute. By reading the lecture notes by Brian Greene, even though I couldn't really understand how one should construct mirror pairs, I found out that one needs to mod out by certain groups such as, for example, Z_5^3 at the end, when one construct mirror pairs. This looked very strange to me, as it seemed to me that this gives rise to the asymmetry between mirror pairs, as one side of mirror pair is moded out version of certain quintic hypersurface, while the other side of mirror pair is un-moded out version of certain quintic hypersurface. However, by reading page 478 of the thick book "Mirror Symmetry," I found out that this point of view of mine was wrong. There is no asymmetry. Here is an excerpt:

"Moreover a curious general symmetry of all conformal theories implies that if we consider an orbifold of a conformal field theory C1 by an abelian group G, denoted by C2=C1/G, then there is an orbifold of the new theory by the same group wich gives back the original theory. C1=C2/G"

Then the authors go on to explain that C/Z_5^4=C/Z_5 as C/Z_5^5=C