The dolman, pelisse or atilla (the jacket had many names through history) was lined with fur and heavily decorated with stiff braided or knottet stitching, both of which provided some cover from glancing lance strikes or sabre cuts - it is possible that the tradition of wearing it slung over your left shoulder was a carry-over from earlier Magyar light cavalry tradition to wear fur the same way.

It makes some sense condering your right-handed enemy would most likely strike you from your left, while you needed your own right arm free to use your own sabre.

That is supposedly why the original Hungarian hussars did it, and the rest of Europe copied the fashion when they copied the Hussar style light cavalry in the mid-18th century.