With lots of help from Andrew May's WebLog, I figured it out. Here are some links:
http://blogs.msdn.com/andrew_may/archive/2006/07/10/SharePointBeta2InfoPathFormsinContentTypes.aspx
http://blogs.msdn.com/andrew_may/archive/2006/06/14/SharePointTermsColumnsFieldsProperties.aspx
http://blogs.msdn.com/andrew_may/archive/2006/06/29/SharePointBeta2XMLDocumentParsers1.aspx (#1 of 5)
My original form library was a mess, so I ended up deleting it and all the form content types so that I was starting from scratch. Then I did these steps:
1. I realized I needed two form libraries, Case Forms and Admin Forms, each of which would need its own set of common promoted fields.
2. Define all of the common fields as site columns.
Site Actions | Site Settings | Galleries | Site Columns.
3. Define a "Case Form" site content type with "Form" as its parent.
Site Actions | Site Settings | Galleries | Site Content Types.
4. Add the site columns you want to the "Case Form" content type.
5. Define a "Admin Form" site content type with "Form" as its parent.
6. Add the site columns you want to the "Admin Form" content type.
7. For each InfoPath form template:
a. Change the Data Connection to point to the appropriate Form Library.
b. Remove all promoted fields (Tools | Form Options | Property Promotion).
c. Save the form template.
d. Publish the form template as a site content type.
e. The first time you publish it, it will be a new content type.
Make sure you select either "Case Form" or "Admin Form" as the
content type that you want to base the form template on.
f. After the first time you publish, it will be an update of an existing type,
which is the previously published form template.
g. For each form field you want to promote:
- Select the form field
- Select the appropriate site column that was created in step #2.
Do not create a new list column! (That's the mistake I made).
That's it. You can use the common columns to sort and filter the forms however you'd like.
Chris