Even and Odd Filters and Quadrature
An even function is symmetric:

$$f(x) = f(-x)$$

An odd function is anti-symmetric:

$$-f(x) = f(-x)$$

A signal in quadrature with another has all its Fourier series components phase-shifted by 90 degrees. The two signals are thus orthogonal.

To get the quadrature signal we apply the Hilbert transform (or Riesz transform in more than 1 dimension)

$$g(x) = \pm \mathcal{H}[f](x)$$

A cosine wave is even and symmetric. A sine wave is odd and anti-symmetric. Therefore if you apply the Hilbert transform to an even signal (cosine Fourier series components) you get an odd signal (sine Fourier series components). If you apply the Hilbert transform again you get the negative original signal. Apply again you get the negative of the original Hilbert transformed signal. Apply again you get the original signal. This is where the "quad" part comes in.

This is useful in signal analysis. Since the Hilbert transform only phase-shifts the Fourier series components, the energy of the signal remains constant. We can thus reconstruct the original signal (e.g. wavelet denoising). Also, given an even filter that responds to peaks and troughs, we can create an odd filter that responds to edges. The pair of filters is called a quadrature filter pair and allows the analysis of signal features. See the analytic signal for more details.