The article "Vector reconstruction from Firing Rates" of Abbott and Salinas
What they mean is that as long as you rotate $C_i$ and $C_j$ by the same amount, the dot product will be the same.  This is rotational invariance because the dot product is invariant to coherent rotation of the relevant vectors.  

In neural terms, the correlation between the activity of two neurons (in a population representing a one dimensional circular variable) is a function of the difference in their preferred orientations and does not depend on the absolute direction of their orientations.