I would look for their interest in the mathematical study of [even and odd functions][1], which have their own wikipedia pages. Odd and even functions have inherent properties (symmetry, limit conditions, sum/product) that simplify their analysis. In turn, as every function can be represented uniquely as a sum of an even and an odd function:

$$ f(x) = \frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$

this property can be used in more complicated contexts. For instance, if $f$ is differentiable and even, the derivative is odd, and *vice versa*. Consequently, this decomposition is useful to analyze differential equations. Now, remember that [Fourier discovered his series by studying and solving the heat equation][2]. So, it is not surprising to see those odd/even symmetries at play with Fourier related bases: [discrete cosine and sine transforms][3], [Hartley transform][4]. The symmetries can be used to reduce complexity and provide faster algorithms.

There is also a more algebraic and statistical derivation. They are detailed from the 1976 papers [Properties of the Eigenvectors of Persymmetric Matrices with Applications to Communication Theory][5] and [Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices][6] by Cantoni and Butler. They observe that many problems (communication theory, signal processing) lend themselves to using eigenvalues and eigenvectors of certain matrices:


1. Information theory, for example discrete time channel equalisation  and maximum likelihood PAM detection
2. Linear system theory, for example, stability of discrete time systems
3. Linear estimation theory (the covariance matrix of a stationary stochastic process belongs to the class), for instance Principal Component Analysis
4. Numerical analysis, for example in the solution of differential equations

Then, there is a nice theorem:

> It is proved that the eigenvectors of a symmetric centrosymmetric
> matrix of order $N$ are either symmetric or skew symmetric, and that
> there are $\lceil N/2 \rceil$ symmetric and $\lfloor N/2 \rfloor$ skew symmetric eigenvectors. Some
> previously known but widely scattered facts about symmetric
> centrosymmetric matrices are presented for completeness. Special cases
> are considered, in particular tridiagonal matrices of both odd and
> even order, for which it is shown that the eigenvectors corresponding
> to the eigenvalues arranged in descending order are alternately
> symmetric and skew symmetric provided the eigenvalues are distinct.

For instance, a basis derived from a covariance matrix for a signal of length $2K$ will have $K$ odd and $K$ even eigenvectors (under mild conditions). Hence, when designing a basis, it comes as natural to have both odd (skew-symmetric) and even vectors. Interestingly, classical orthogonal dyadic discrete wavelets do not exhibit such a symmetry, inspiring other designs like multi-band wavelets or lapped orthogonal transforms (LOT), that are real, have finite support and odd/even symmetries.


  [1]: https://en.wikipedia.org/wiki/Even_and_odd_functions
  [2]: https://en.wikipedia.org/wiki/Heat_equation#Solving_the_heat_equation_using_Fourier_series
  [3]: https://dsp.stackexchange.com/a/41624/15892
  [4]: https://dsp.stackexchange.com/q/34677/15892
  [5]: https://doi.org/10.1109/TCOM.1976.1093391
  [6]: https://doi.org/10.1016/0024-3795(76)90101-4