The person who created the open-invite game has created a "set-piece" game, where the pieces have been arranged in positions that represent something other than the regulation starting position. Perhaps it represents the first few moves of some book opening that they wish to practice on.
If you look at the bottom of the open-invite page there is an icon that represents a set piece game, and that icon should also appear on the right side of the line where you found the game. You can click on the icon there and see what the position is before you accept that game. The person who created it may have also included information about what the position represents, as well.
It is not a rated game, but I believe it would still count as a win or loss in your statistics.
I hope this helped.