One sided frequency spectrum (Matlab vs. Origin)
The indexing is a matter of convention.  "Natural" is zero centered, "Computer implementation" is zero based.  

We are having a big discussion right now over several questions on why your $c_{-5}$ should actually be considered as $(c_{-5} + $c_{5})/2$.  You'll find part of the discussion here Convergence of periodic sinc interpolation and the links to the rest are embedded within.

The "fluffy cloud" drawings show what happens when you consider the upper half (from a zero based perspective) as positive frequencies.  How to get Fourier coefficients to draw any shape using DFT?

If I have a circle, and I check your location every minute and it seems you have moved a quarter circle counter-clockwise each time, how do I know you haven't actually moved three quarters of the circle clockwise?

Answer:  You don't.  The convention decides which interpretation to use.