Here's a bit of followup.
I found something that turns off gzip compression through .htaccess, which has solved my problem.
mod_gzip_on no
I will admit that we've made a few other changes to the server set up, so it may not quite be as simple as that, but from what I see, this is what's fixed it for me.
For reference, we upgraded PHP from 4.4.7 to 5.2.0 through WHM, which means that Apache was rebuilt in the process. We're also running Apache 1.3.