Strangely, the gigya shortcode is missing from that doc (perhaps because it’s not as easy to use as, say, the youtube shortcode). Gigya is the shortcode that will show up in your html editor when you use the auto-post-to-WP button that some sites with videos or other flash objects provide. I first got acquainted with it when I experimented with the Mixpod audio players.
Using this shortcode, you can insert any video that provides a flash embed code (not directly allowed in wp.com). You copy the URL and the dimensions of the video from the embed code, and you transform them into a working shortcode this way:
Paste into html post or page editor. In contrast to some other shortcodes, the gigya shortcode works in text widgets too; of course you’ll need to adjust the dimensions – see my post on sidebar width.
For YouTube videos in particular, you have to click “Embed” then tick the option “Use old embed code”.
If the embed code is like this:
Turn it to this:
Same thing if there’s a string of variables appended to the video URL after a questionmark.
For Kickstarter videos in particular, see separate post.