Walter Wink, of Spirituality & Health, has it right I think.
Here is my analysis, written before reading his.
An alternate interpretation of the scripture.
At this time, a thief would have a hand chopped off. People ate from a common plate of food in the middle of the table. One used one hand to wipe after defecating and the OTHER hand to take food from the common plate. Thus a man with one hand was not welcome to eat with others. Usually, the eating hand was the right hand.
Thus if one were struck on the right cheek by the back of someone's right hand, it was an insult and a provocation.
What happens if one turns the OTHER cheek, presumably to be struck with the OTHER hand?
I believe if you were struck AGAIN with the OTHER hand, it would be a grievous insult and a major provocation.
This leads me to conclude that the gesture of turning the other cheek is a challenge to the striker. It says, "Look you jerk, you hit me. I'm raising the stakes. If you hit me again with the OTHER hand, you will demonstrate that you are a major jerk, unworthy of any admiration or status."
This is, thus, a non-violent act intended to SHAME the other person, perhaps even into backing down and apologizing. It is good social policy since it leaves everyone involved with their teeth and eyes intact.
Does that make sense to you?
Similarly, the act of offering your shirt as well as your coat leaves you naked in the street. This was a big deal in those days, and even today. Again it challenges and shames the other person.