Recoil starter help on WB 820

I finished rebuilding the West Bend 820 on my Chrysler Sno Runner. Both the recoil starters I have do not engage the "Cup", which has many gouges in it. When looking for a replacement cup I saw one advertised as a right hand "Cup". Is there a difference in starter cups on WB right and left engines. This engine has the pull start on the left with the carb facing forward and has a left hand thread on the crank.
Thanks in advance.
As it turns out, both of the recoils I had were assembled wrong. After trying many combinations I ended up getting the spring and washer combo correct and they both now work.