My take is only slightly different. Remember the context is not just a backhanded slap. It is likely (I hold this interpretation loosely--I always reserve the right to change my mind!) he is speaking of a Roman soldier backhanding a Jewish person.

I may assume too much, but I imagine a hard, backhanded strike from a soldier that sends someone sprawling. Instead of staying put, the victim stands back up, faces the opponent and looking him in the eye turns the left cheek.

Similar scenario; and it is definitely not a milquetoast response. It is, in fact, a risky response (the soldier might run you through with his gladius). It is confrontational.