fftshift in MATLAB with even number of data points in double sided spectrum
If the size $N$ of the DFT is even, only one "extremal" point (after fftshift) is Nyquist.  If $N$ is even, you cannot have an arbitary $c_{-5}$ and $c_5$.  They must add to be whatever your $c_{-5}$ term is.

If the input to the DFT is purely real and if $N$ is even, consider the Nyquist point, at $c_{N/2}$ to be split in half.  One half is the negative-frequency component and the other half is the positive-frequency component.  Since the negative and positive frequency components must be complex conjugates of each other, and since $c_{-N/2}=c_{N/2}$ after splitting, that means that $c_{N/2}$ must be purely real just as $c_0$ must also be real.