If photons don't interact directly, how can electromagnetic waves interfere? I know that photons can scatter via higher order mechanisms, but not directly. Does those mechanisms explain the classical phenomenon of wave interference?
Note carefully Nick's comment. Suppose I send two plane EM waves on some collision course so they interfere. The waves will pass through the region where they meet, generating some interference pattern in that region, then they will exit that region and continue on their separate ways unchanged. In other words neither the energy nor the momentum of the waves has change as a result of their crossing i.e. they have not interacted in the usual QM sense of applying some force to each other.
The interference is not the result of the two EM waves interacting with each other, it's the result of we, the observer, interacting separately with both fields. So if the effect of our interaction with field A is equal and opposite to the effect of our interaction with field B the net effect on us is zero and we get a zero in the interference pattern. Likewise if the two interactions reinforce each other we get a maximum.
There is no need for high order mechanism. It is simply because a single photon can interfere with itself. If you remember the double slit experiment, they are indeed looking for a single photon passing through a slit and interfere with itself.
Now if, instead we have billions of billions photons, the same single photon interference still happen individually, but now you can see it macroscopically as a "classical electromagnetic wave" interference. Though most of them are still interfere with itself.
The scattering process you mentioned is less important than the above process.