Thanks for the great ideas everyone. There were a few suggestions that I thought were probably more techniques for embellishing or decorating cards rather than a card foundation, so I've left those off this list but hope to get to them another time. Also, I've noticed that some cards have more than one name, so I'll list them under both as I come across them (eg: wobble cards are also known as rocker cards, flip flop cards are also swing card, etc.).
These are the ideas I'm going to add, along with the people who first suggested them:
CardArTart - never ending card
mforquer - double side step card
Jacee - centre step card
CraftyGirl - funky fold card, tilt card, exploding card, pyramid fold card, s-fold card, diamond fold card, triangle tri-fold card, double slider card, slider window card, trellis card, magic window card, flip flop card, pop-up twirl card, free standing pop-up card, double diamond card, double tri-shutter card
Jessica - hidden message card
Theresa F - secret message card, venetian blind card