I never thought about mounting only on one side or that it can look silly. I too tend to mount on the left but that is because I carry my bike on my right side so I don't get drivetrain grease on me. I hate it when I get chain marks on me or my clothes. (I have to carry my bike down my long, gravel driveway. Too many thorns hiding in the rocks I discovered 3 flats later.)

Now I will have to try mounting on the right side next ride just to see how it feels.