Normally the way the preference works is that you "give up" your shares in exchange for being paid back, as if you had initially given the company a loan instea

Normally the way the preference works is that you "give up" your shares in exchange for being paid back, as if you had initially given the company a loan instead of bought equity. E.g., you invest $1M, company sells for $15M, and you want to be able to get $2M (2x) of the $15M in exchange for your investment. With regular preferred shares, you get paid your $2M and then that's it, your initial $1M is paid back. With participating preferred, you get your $2M, but then act as if you still had the equity that you bought with the $1M (even though you basically already got paid back for it). So you get $2M + whatever your cut of the remaining $13M is.