To flip a YouTube watch page, go to the watch page you want to flip and copy the exact URL from the labeled box on the right. (Be sure it's the URL box and not the embed box.) This will get you the exact URL without any extraneous code that could interfere with this trick.
Next, paste your URL into your browser window and add &flip=1 to the end of it with no spaces. That's it.
So if your URL used to be: http://www.youtube.com/watch?v=8HBVIU0wevg
it should now be: http://www.youtube.com/watch?v=8HBVIU0wevg&flip=1
Your entire watch page should now be upside down. If you copy the code from your browser window, you can link other users to your flipped video watch page.
However, as an undocumented YouTube hack, this could stop working at any time.
Update: Sure enough, it looks like this hack no longer works. Perhaps they'll pull it out again for April Fool's next year.