The obvious ones are 0 and $e^{x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?

The following is discussed in a little more detail on pages 337339 of Frank Jones's book "Lebesgue Integration on Euclidean Space" (and many other places as well). Normalize the Fourier transform so that it is a unitary operator $T$ on $L^2(\mathbb{R})$. One can then check that $T^4=1$. The eigenvalues are thus $1$, $i$, $1$, and $i$. For $a$ one of these eigenvalues, denote by $M_a$ the corresponding eigenspace. It turns out then that $L^2(\mathbb{R})$ is the direct sum of these $4$ eigenspaces! In fact, this is easy linear algebra. Consider $f \in L^2(\mathbb{R})$. We want to find $f_a \in M_a$ for each of the eigenvalues such that $f = f_1 + f_{1} + f_{i} + f_{i}$. Using the fact that $T^4 = 1$, we obtain the following 4 equations in 4 unknowns: $f = f_1 + f_{1} + f_{i} + f_{i}$ $T(f) = f_1  f_{1} +i f_{i} i f_{i}$ $T^2(f) = f_1 + f_{1}  f_{i}  f_{i}$ $T^3(f) = f_1  f_{1} i f_{i} +i f_{i}$ Solving these four equations yields the corresponding projection operators. As an example, for $f \in L^2(\mathbb{R})$, we get that $\frac{1}{4}(f + T(f) + T^2(f) + T^3(f))$ is a fixed point for $T$. 


Following on a little from Andy's comment, Hermite polynomials (multiplied by a Gaussian factor) give a basis of eigenvectors for the FT as an operator on $L^2({\mathbb R})$ 


A very important fixed point of the Fourier transform that isn't in $L^2$ is the Dirac comb distribution, informally $$D(x) = \sum_{n\in Z} \delta(xn),$$ or more properly, defined by its pairing on smooth functions of sufficient decay by $$\langle D, f\rangle = \sum_{n\in Z} f(n).$$ The fact that $D$ is equal to its Fourier transform is really just the Poisson summation formula. (I wrote an argument explaining why $D$ should be its own Fourier transform in an answer to another question: Truth of the Poisson summation formula) 


$\bf{1.}$ A more complete list of particular selfreciprocal Fourier functions of the first kind, i.e. eigenfunctions of the cosine Fourier transform $\sqrt{\frac{2}{\pi}}\int_0^\infty f(x)\cos ax dx=f(a)$: $1.$ $\displaystyle e^{x^2/2}$ (more generally $e^{x^2/2}H_{2n}(x)$, $H_n$ is Hermite polynomial) $2.$ $\displaystyle \frac{1}{\sqrt{x}}$ $\qquad$ $3.$ $\displaystyle\frac{1}{\cosh\sqrt{\frac{\pi}{2}}x}$ $\qquad$ $4.$ $\displaystyle \frac{\cosh \frac{\sqrt{\pi}x}{2}}{\cosh \sqrt{\pi}x}$ $\qquad$$5.$ $\displaystyle\frac{1}{1+2\cosh \left(\sqrt{\frac{2\pi}{3}}x\right)}$ $6.$ $\displaystyle \frac{\cosh\frac{\sqrt{3\pi}x}{2}}{2\cosh \left( 2\sqrt{\frac{\pi}{3}} x\right)1}$ $\qquad$ $7.$ $\displaystyle \frac{\cosh\left(\sqrt{\frac{3\pi}{2}}x\right)}{\cosh (\sqrt{2\pi}x)\cos(\sqrt{3}\pi)}$ $\qquad$ $8.$ $\displaystyle \cos\left(\frac{x^2}{2}\frac{\pi}{8}\right) $ $9.$ $\displaystyle\frac{\cos \frac{x^2}{2}+\sin \frac{x^2}{2}}{\cosh\sqrt{\frac{\pi}{2}}x}$ $\qquad$ $10.$ $\displaystyle \sqrt{x}J_{\frac{1}{4}}\left(\frac{x^2}{2}\right)$ $\qquad$ $11.$ $\displaystyle \frac{\sqrt[4]{a}\ K_{\frac{1}{4}}\left(a\sqrt{x^2+a^2}\right)}{(x^2+a^2)^{\frac{1}{8}}}$ $12.$ $\displaystyle \frac{x e^{\beta\sqrt{x^2+\beta^2}}}{\sqrt{x^2+\beta^2}\sqrt{\sqrt{x^2+\beta^2}\beta}}$$\qquad$ $13.$ $\displaystyle \psi\left(1+\frac{x}{\sqrt{2\pi}}\right)\ln\frac{x}{\sqrt{2\pi}}$, $\ \psi$ is digamma function. Examples $15,810$ are from the chapter about selfreciprocal functions in Titschmarsh's book "Introduction to the theory of Fourier transform". Examples $11$ and $12$ can be found in Gradsteyn and Ryzhik. Examples $6$ and $7$ are from this question What are all functions of the form $\frac{\cosh(\alpha x)}{\cosh x+c}$ selfreciprocal under Fourier transform?. Some other selfreciprocal functions composed of hyperbolic functions are given in Bryden Cais's paper On the transformation of infinite series. Discussion of $13$ can be found in Berndt's article. $\bf{2.}$ Selfreciprocal Fourier functions of the second kind, i.e. eigenfunctions of the sine Fourier transform $\sqrt{\frac{2}{\pi}}\int_0^\infty f(x)\sin ax dx=f(a)$: $1.$ $\displaystyle \frac{1}{\sqrt{x}}$ $\qquad$ $2.$ $\displaystyle xe^{x^2/2}$ (and more generally $e^{x^2/2}H_{2n+1}(x)$) $3.$ $\displaystyle \frac{1}{e^{\sqrt{2\pi}x}1}\frac{1}{\sqrt{2\pi}x}$ $\qquad$ $4.$ $\displaystyle \frac{\sinh \frac{\sqrt{\pi}x}{2}}{\cosh \sqrt{\pi}x}$ $\qquad$ $5.$ $\displaystyle \frac{\sinh\sqrt{\frac{\pi}{6}}x}{2\cosh \left(\sqrt{\frac{2\pi}{3}}x\right)1}$ $6.$ $\displaystyle \frac{\sinh(\sqrt{\pi}x)}{\cosh \sqrt{2\pi} x\cos(\sqrt{2}\pi)}$ $\qquad$ $7.$ $\displaystyle \frac{\sin \frac{x^2}{2}}{\sinh\sqrt{\frac{\pi}{2}}x}$ $\qquad$ $8.$ $\displaystyle \frac{xK_{\frac{3}{4}}\left(a\sqrt{x^2+a^2}\right)}{(x^2+a^2)^{\frac{3}{8}}}$ $9.$ $\displaystyle \frac{x e^{\beta\sqrt{x^2+\beta^2}}}{\sqrt{x^2+\beta^2}\sqrt{\sqrt{x^2+\beta^2}+\beta}}$$\qquad$ $10.$ $\displaystyle \sqrt{x}J_{\frac{1}{4}}\left(\frac{x^2}{2}\right)$$\qquad$ $11.$ $\displaystyle e^{\frac{x^2}{4}}I_{0}\left(\frac{x^2}{4}\right)$ $12.$ $\displaystyle \sin\left(\frac{3\pi}{8}+\frac{x^2}{4}\right)J_{0}\left(\frac{x^2}{4}\right) $$\qquad$ $13.$ $\displaystyle \frac{\sinh \sqrt{\frac{2\pi}{3}}x}{\cosh \sqrt{\frac{3\pi}{2}}x}$ Examples $15,7$ can be found in Titschmarsh's book cited above. $812$ can be found in Gradsteyn and Ryzhik. $13$ is from Bryden Cais, On the transformation of infinite series, where more functions of this kind are given. 

