Intuitively, why is the Gaussian the Fourier transform of itself?

The generalization of this phenomenon, from a probabilistic standpoint, is the Wiener-Askey Polynomial Chaos.

In general, there is a connection between orthogonal polynomial families in the Askey scheme and probability distribution/mass functions.

Orthogonality of these polynomials can be shown in an inner product space using a weighting function -- a weight function that typically happens to be, within a scale factor, the pdf/pmf of some distribution.

In other words, we can use these orthogonal polynomials as a basis for a series expansion of a random variable:

$$z = \sum_{i=0}^\infty z_i \Phi_i(\zeta).$$

The random variable $\zeta$ belongs to a distribution we choose, and the orthogonal polynomial family to which $\Phi$ belongs follows from this choice.

The deterministic coefficients $z_i$ can be computed easily by using Galerkin's method.

So, yes. There is a very deep connection in this regard, and it is extremely powerful, particularly in engineering applications. Strangely, many mathematicians do not know this relationship!

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See also: http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA460654 and the Cameron-Martin Theorem.