How to get back a declined Outlook invitation

The invite should still be in the "Deleted Items" Folder. If you go there, you will see the Accept / Decline options again. Click the correct one.

I went to the sender's calendar and clicked the "copy to calendar" button in the ribbon at the top (as described on this page). It prompted me to accept the invite - easy.

Periodically I accidentally delete a request before replying, and I've always had success with just dragging it out of Deleted Items back into the Inbox, then opening it and sending the reply of my choice. I'm not sure I've ever tried changing a response once sent, but all the response buttons do display, so it'd be worth a try.
Periodically I accidentally delete a request before replying, and I've always had success with just dragging it out of Deleted Items back into the Inbox, then opening it and sending the reply of my choice. I'm not sure I've ever tried changing a response once sent, but all the response buttons do display, so it'd be worth a try.

With a regular email you can go to the Sent Items folder and click Recall This Message and Exchange will attempt to recall it. That option does not seem to be available for an invitation to a metting. I also looked at Matthew's answer and my version of Outlook (2003) shows only the declined invite in the deleted folder. I do not have the option to change my response. I think all you can do is ask the sender to invite you again.