Admittedly I've only looked at modern (electronic) AF 35mm cameras, but they all go in alternating directions. Shutter is closed with both curtains at one side (actually, top or bottom) of the frame and an exposure is made by moving one curtain then the other, leaving the shutter again closed by with the curtains on the opposite side. It then does the opposite for the next frame.

I've little idea how it works mechanically but see no good reason for it not to be symmetric - there would merely be two tensioning mechanisms and they are activated alternately when winding on.