Thanks again for your help, I think I've finally figured this out ...
The points that I'm seeing are offsets of the starting point. Here's the xml that gets generated for my polygon:
<object id="8" x="108" y="460">
<polygon points="0,0 36,-32 116,-29.3333 129.333,-45.3333 513.333,-48 681.333,116 678.667,524 640,522.667 628,505.333 586.667,506.667 586.667,518.667 568,525.333 568,556 16,557.333 5.33333,530.667 -52,532 -52,66.6667 -1.33333,25.3333"/>
When I take the (x, y) coordinate and add it to the points in the list, I get the correct positions. I tried both deleting/recreating the polygon and creating an entirely new map; in both cases, this is how the xml comes out. I'm not entirely sure why it gets generated this way, but at least I know how to deal with it now.
Note that per the original question here, Extended.Tiled collision can be (basically) handled by creating a Polygon object from the map points and checking it against the target position.