What is it that is "symmetry-adapted" about symmetry-adapted perturbation theory? I think this is a fairly straightforward question which I should know the answer to, but surprisingly cannot find a discussion of online.

Symmetry-adapted perturbation theory (SAPT) is a fairly common method to describe intermolecular interactions using physically meaningful terms (electrostatics, exchange, induction, dispersion), where the interaction is built up. That is, SAPT calculates the interactions directly as opposed to finding the interaction strength by subtraction.

So, quite simply, in SAPT, what does the "symmetry-adapted" specifically mean, and why is it necessary to specify this? 

In other words, is there a non-symmetry-adapted perturbation theory which can be used for intermolecular interactions?

Finally, how, if at all, does this symmetry-adaptation relate to the kind we use when constructing molecular orbitals using group theory?
The "symmetry" in symmetry-adapted perturbation theory refers to the anti-symmetry of the wave function with respect to electron exchange:

$$\Psi(\mathbf{r}_1,\mathbf{r}_2) = -\Psi(\mathbf{r}_2,\mathbf{r}_1) $$

The most usual way to enforce antisymmetry in electronic structure theory is to enforce orthogonality between spin-orbitals.  So SAPT is intermolecular perturbation theory that includes terms that depend on the overlap between orbitals in different molecules.

Intermolecular perturbation theory gives rise to concepts like eletrostatic interactions, polarization, and dispersion.  With SAPT you then get additional concepts like exchange repulsion (aka steric or Pauli repulsion).