I think the turn the other check is generally misunderstood because we fail to understand the culture and times Jesus was in . .
Then, if someone hit you (imagine someone slapping you on the face . . it is their palm, right?) they it you with the flat of their palm. If you turned the other check (literally) and they tried to strike you again, they would be forced to backhand you . . now, this demeaned them if you forced them to do so . . you would be preventing them from hitting you again and demonstrating the error of their actions plainly without a word or fight.
Same with the admonishion of carrying a load the extra mile . . the Romans were able to force someone to carry something, but only for a mile . . if they made someone carry it for farther then that they could get in serious trouble . . so, carrying it for another mile was something that would put the soldier in a dreadful position . . if you carried it beyond to proscribed limit, you were turning the tables on the soldier and where he initially forced you to carry something against your will, now you would be putting him in a position of begging you to give it back . .very humbling to the soldier, and an action that would make him think twice of forcing someone to his will again . .
The one about the cloak also is the same . . if you were to give them your undergarment, then you wold be essentially naked . . imagine standing in public when soemone made you give them your cloak . . if you took off all your clothes, at least down to your underwear and handed them over as well, how do you think it would make the other person look to everyone around them?
This is what Christ was speaking of . . today these analogies do not hold. . because we are not in the same culture or times . .
What Christ was saying is respond to them in such a way that gently and lovingly shows them their error . .
Peace in Him!