Bagpuss The Gnome:
I don't remember seeing it in the Satipatthana Sutta or even in the parts of the Majjhima Nikaya i have read so i presume it's a later practice in Mahayana?
Actually, I think self inquiry is not a buddhist practice per se, although it gets mixed in a lot, especially in pragmatic dharma circles. As far as I can tell, the source of this practice is the advaita vedanta, which, if I'm not mistaken, goes back to early brahmanism. The central idea there seems to be that there is this eternal unchanging ever-present all-encompassing primordial awareness/consciousness (the atman) , and you are it, and all phenomena arise in this awareness without ever changing it, touching it, threatening it, etc. The core elements of this system seem to have already existed in the buddha's lifetime and his teaching on anatta (or an-atman) can be read as a dircet reaction to this point of view.
Regarding books, I can recommend taking a look at "the direct path" by Greg Goode, which is basically a series of experience-based contemplations, where you work through all your experience bit by bit and look for the self. This practice can be interesting, but it's analysis-heavy and doesn't have much to do with actual meditation (as in noting or concentration or metta etc.). The outcome is also predetermined and therefore not very surprising. So if the underlying philosophical axiomatics (eternal big self) don't bother you, you could give it a try. (Greg Goode also has some videos on youtube where you can get a good impression of his approach.)