the_content() returning null in one category only, even though there is content

Is there any way to output the raw, unfiltered post HTML to find out
whether this is a rogue filter deleting all the content in that
specific category for some weird reason?

Yes there is, you can use $post->post_content. Try adding something like:

echo '<pre>'.$post->post_content.'</pre>';

just before the_content() call in your template file.

What else could cause a wholesale deletion of content in one category
only?

Since you say that you can see the content in the back end there is no deletion and most probably has something to do with a filter hook, hooked with the_content hook.

try deactivating your plugins and reactivating theme one by one to see which one is causing it. And if that is not it then try switching to the default theme see if that is the cause.