There is a different conceptual way to think about this. Imagine that you have a necklace made up of beads of different weights. Imagine also that you have a wheel with a circumference the same length as the necklace. Now, attach the necklace at the 3 o'clock position on the wheel and wrap it counter-clockwise around the wheel. Next, wrap the necklace in a clockwise direction. By symmetry, it is obvious that the center of masses in the two cases are going to be mirror imaged on the horizontal center line of the wheel. The same is true if you stretch the necklace by a factor of two and wrap it twice around the wheel. Same for 3, same for 4, etc. If there are $N$ evenly spaced beads, and you are stretching the necklace $N-1$ times, it is easy to see the beads end up as if you were wrapping the necklace in the reverse direction once.
This is precisely how a $1/N$ normalized DFT of a real valued signal works and one way to look at what it means. (See my blog article DFT Graphical Interpretation: Centroids of Weighted Roots of Unity for a more mathematical description.)
The center of mass (bin value) will be in the mirror image position (complex conjugate) when you wrap the necklace (DFT of signal) in the reverse (negative) direction.
$$ X[k] = X^*[-k] $$
So, this pertains to real valued signals. Suppose you have $M$ bins, and assume that $M$ is even, then bins $1$ through $M/2-1$ are the complex conjugates of bins $M-1$ through $M/2+1$, so those are your redundant spectral coefficients. Furthermore, the DC bin ($0$) and the Nyquist bin ($M/2$) have to be real valued, that is the imaginary component is zero.