I was asked about a few millinery related things in the past migraine infused week. Here is one quick look at symmetry and asymmetry in straw hats of the Civil War era. (Please note: This does not break down the location of trim by yearly or seasonal fashion.)
Image Sources:
MET, MFA, Peterson’s Magazine, assorted pins on Pinterest.
If I Had My Own Blue Box: 
Intertesting – I like both, but I wonder if we might characterize the asymetrical hats as a female form of a hat tipped over one eye or to the side on a man? A little more flirtatious perhaps?