At the time of Jesus, striking someone deemed to be of a lower class with the back of the hand was used to assert authority and dominance. If the persecuted person "turned the other cheek," the discipliner was faced with a dilemma. The left hand was used for unclean purposes, so a back-hand strike on the opposite cheek would not be performed. The other alternative would be a slap with the open hand as a challenge or to punch the person, but this was seen as a statement of equality. Thus, by turning the other cheek the persecuted was in effect demanding equality.
By handing over one's cloak in addition to one's tunic, the debtor has essentially given the shirt off their back, a situation directly forbidden by Jewish Law as stated in Deuteronomy 24: 10-13:
When you make your neighbor a loan of any sort, you shall not enter his house to take his pledge. You shall remain outside, and the man to whom you make the loan shall bring the pledge out to you. If he is a poor man, you shall not sleep with his pledge. When the sun goes down you shall surely return the pledge to him, that he may sleep in his cloak and bless you; and it will be righteousness for you before the LORD your God.
By giving the lender the cloak as well the debtor was reduced to nakedness. Public nudity was viewed as bringing shame on the viewer, not the naked, as evidenced in Genesis 9: 20-27:
Noah was the first tiller of the soil. He planted a vineyard; and he drank of the wine, and became drunk, and lay uncovered in his tent. And Ham, the father of Canaan, saw the nakedness of his father, and told his two brothers outside. Then Shem and Japheth took a garment, laid it upon both their shoulders, and walked backward and covered the nakedness of their father; their faces were turned away, and they did not see their father's nakedness.
The succeeding verse from the Sermon on the Mount can similarly be seen as a method for making the oppressor break the law.
The commonly invoked Roman law of Angaria allowed the Roman authorities to demand that inhabitants of occupied territories carry messages and equipment the distance of one mile post, but prohibited forcing an individual to go further than a single mile, at the risk of suffering disciplinary actions. In this example, the nonviolent interpretation sees Jesus as placing criticism on an unjust and hated Roman law as well as clarifying the teaching to extend beyond Jewish law.
 As a side effect this may also have afforded the early followers a longer time to missionary to the soldier and or cause the soldier not to seek followers of Jesus to carry his equipment in the future so as not to be bothered with their proselytizing.