So what do you see if you open one of these messages in non-Apple software?

Thanks for your reply! If you open an email in Thunderbird with a pdf you will see an attachment is available at the bottom and you can click to view the name of the attached pdf. You can then double click the attached pdf thus opening with Adobe Reader in a separate window.

Apple Mail actually appends the message as attachment. It only has built in a way to display the attachment inline, something that Thunderbird doesn't. I think that if you add it as attachment with Thunderbird, and display the mail in Apple Mail, it will also show the contents (at least when the PDF has only one page).