I am not sure if this is in the spirit of what you are asking but it is a natural explanation of why the Schwartz space has this property. The setting is the following. Suppose that $A$ is an unbounded, positive self-adjoint operator on Hilbert space. Then the spaces $D(A^\alpha)$ and $\bigcap D(A^n)$ are in a natural way Hilbert resp. Fréchet spaces (the symbol $D$ denotes domain of definition). The relevance to your question is that if one chooses for $A$ the standard one-dimensional Schrödinger operator, then one obtains the Schwartz spaces and associated Sobolev-type spaces. One can also dualise the construction to get suitable spaces of distributions, with the tempered distributions as the limit case.
The connection with the Fourier transform comes from the fact that this operator commutes with the Schrödinger operator and this implies that it is a self-mapping of these spaces. The same thing will happen with any $A$ which commutes with the F.T. and this will provide a plethora of spaces of test functions, distribution spaces and Sobolev spaces with the property you desire.