Discrete Fourier transform symmetry
Even &amp; odd refer to the symmetry around $n = 0$.

Even means $x[n] = x[-n]$; you can get the part for $n &lt; 0$ by simply mirroring the part for $n &gt; 0$ at the $n=0$ line.

Odd means $x[n] = -x[-n]$; you can get the part for $n &lt; 0$ by simply mirroring the part for $n &gt; 0$ at the $n=0$ line and multiplying it by $-1$.

A cosine wave is even, sine wave is odd. 

These are all just special cases of the general symmetry that says 


  if it's real in one domain, it's conjugate symmetric in the other.


Conjugate symmetric means that the real part is even and the imaginary part is odd. Most people know that a real time domain signal as a conjugate symmetric spectrum, but it also goers the other way around: a conjugate symmetric time domain signal has a real valued spectrum.