Where things can get a little bit confusing is that there can be up to four rear derailleur variants to choose from in a particular series (the M770 XT series referred to above is one such case. M590 Deore is the other that springs to mind). M770 is the reverse-sprung "Rapid Rise derailleur, which Shimano was fond of for a while. M771 is the traditional design. M772 is the "Shadow" type, which has a much lower profile that tucks it in tighter under the cassette and better pivot stops to greatly reduce chain slap. These three are compatible with any 7,8 or 9-speed setup. M773 is the "Dyna-Sys" 10-speed variant, which has a different leverage ratio, so it's only compatible with other Dyna-Sys shifters.
Just ride the bloody thing and enjoy yourself!