First, on the server side, you need to enable HTTP compression and make sure all of the options work for your web services. In IIS 6 the GUI for configuring a web site does provide the ability to enable HTTP compression (it's under the Internet Information Service → Web Sites → Properties (right-click) → Service tab), but there is no way to set which file extensions are compressed. Microsoft has a page that explains how to set this stuff via the command line, but it doesn't seem to work quite right. I searched a bit and found some good explanations, which eventually solved my problem.
The only change I made to my C:\WINDOWS\system32\inetsrv\MetaBase.xml file was to make the HcScriptFileExtensions entries newline delimited instead of space delimited. The same goes for HcFileExtensions as well. Take care that you fix these for both deflate and gzip.
So, I now have aspx and asmx extensions configured for compression and when I hit a page from a browser it is downloaded compressed. Woohoo!
But, when my client program that uses my web services (asmx pages) hit the same pages things were not compressed. Boohoo!
The key is that if you're using automatically generated Web References that extend SoapHttpClientProtocol you need to set the EnableDecompression property to true. E.g.:
using(MyWebService proxy = new MyWebService()) {
proxy.EnableDecompression = true;
...
One other thing... It's somewhat annoying that there's no easier way to determine if compression is working. Using a packet sniffer will work, but it can be a pain to install one, especially since you don't need most of its functionality. The way I checked this stuff was to examine the IIS logs and compare the download size of known pages before and after changing various settings. HTTP compression seems to reduce web text to about 25% of its original size. One annoying thing is that the IIS web log file is appended in a way such that the results are delayed by a minute or so. Also, in a live system this file can grow very quickly, so trying to do quick checks in it can be difficult.