Here is a graphic from North Sails that might help illustrate the differences between the different downwind sails. The “A” sails are asymmetrics (It looks like this is what Jon has). The “S”s are the symmetrics. Note that the odd numbers are reachers and the even numbers are runners. On the right, the “G” sails are the North Gennekers and the “C”s are the coded sails. Note that the coded sails are used for reaching and beating, not running. After-all, the North Code versions have a wire luff stitched into them. Incidentally, on the family Catalina, I have a G2 and S2 on board. I can easily single or double hand the Genneker, but with only one spinnaker pole, the symmetric is only used when I have enough crew so someone can act as bowman. The G2 is a little frustrating insomuch as it has no performance in very light airs and unfortunately, an A1.5 is not in the budget.
This is what our North G2 Genneker looks like (we call her “Pinky”… You got to be real secure in your masculinity to fly something made with neon pink.)