There is one part of the animal which is eaten, but rarely. That is the spleen. The spleen is part of the circulatory system and its main function is to filter the blood and regulate the amount of red blood cells - if there is a shortage, it can produce more; if there is an excess it can kill some off. The spleen lies next to the left kidney in most vertebrates but is long and narrow. Straightened out, the one in the picturebelow is 30 cm / 1 foot long.
Pig's Spleen
It has a similar texture to the relevant animal's liver.
Fergus Henderson, nose-to-tail eating enthusiast and populiser, chef-owner of London's famous St John restaurant has a recipe for rolled pig's spleen here. It is a typically simple, yet flavourful example of his no waste cooking style. The compendium edition of his two nose-to-tail books (eG-friendly Amazon.com link) is a must-have for anyone seriously interested in the concept and the practice.
Another interesting recipe that I came across is this for pig's spleen stuffed with minced pork.