Which way does this ring go on the left crank of this Shimano deore hollow tech external bottom bracket crank?
The side that you say has been rubbed silver by friction is like that on a new ring (I've got two, bought from SJS cycles). From what I can tell from other posts it's intended to stop water and crud getting into the bearing. That would suggest that the non-silver side, the ridged side, should go next to the bearing because this is like an oil seal (although there appears to be no rotational movement between the two). The silver side goes into the recess on the crank arm. The washer is a tightish (waterproof) fit on the spindle so together this should stop water getting into the bearing from the spline on the crank arm. 

Just to clarify, there is probably no rotational movement between the crank arm and the face of the bearing - the inner part of the bearing appears to be free to rotate with the arm and the spindle. However, there must be a seal within the bearing to seal the rotating inner part from the fixed outer part.