Philip,
It works when he sends you a meeting request because his Outlook client is creating the MIME stream, independent of the server, and sending it. Once it has reached your Exchange 2003 server, it is placed in the store in it's native format until your MAPI client asks for it, then Exchange parses and converts it. Unless it's converted, scanned and altered along the way to your exchange server, it renders as a rich MR since Exchange can parse a text/calendar body part.
Now when you send out a MR, the Exchange server has to create the MIME stream and has rules to follow based on what the admins have setup There are several places where admins have control of what leaves the org both at the directory (recipient a contact in the GAL or in your contacts folder in Outlook) and at the SMTP level (Text, HTML, both or TNEF). It's most likely that the admins have forced plain text messages on outbound messages from your org and is why your recipient is receiving the MR in plain text. Have your recipient pull down the message with Outlook Express and look at the MIME stream (props on the message). Outlook uses the same engine as Outlook Express to receive mail via POP/IMAP client settings, make sure in Outlook Express you tell it to leave a copy of the message on the server so Outlook2003 will then receive the mail as well when you are done.
It's also possible, but not likely, the ISP server is scanning the inbound SMTP mail, which they all should do for spam, viruses and such, and removing the body parts it doesn't have in it's safe list.
At last - I'd ask you that if this does not help you resolve the issue, to open up the Support case with our Support Services. This will take some troubleshooting and verification of several config settings...