C and T switch in the half court based on L's rotation. The location of the ball does not determine whether or not L rotates it is one factor in deciding if and when to rotate. In other words just because the ball is on C's side of the floor does not mean that C become T.
You are correct about the last part of your post. They either become new L or remain C in transition based on what position they are in when the ball tranisitions the other way.
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