I have a 2013 Shimano XT rear derailleur. It is part of a 2x10 drivetrain. How can I find out if it is a medium or a long cage one? The picture below shows the derailleur.
The picture is a RD-M780-SGS long cage.
Shimano have three codes for rear derailleur length:
I'm not aware of where this is printed on the RD though so not so helpful.
However Shimano only have one non-clutched XT Dyna-Sys (10 speed) RD the RD-M780-SGS (long cage 43t capacity). The clutched (shadow+) RDs come in GS (medium 35t cap) and SGS (long 43t cap).
Saint derailleurs come in SS (short 23t cap with cassette extender) and GS (medium 37t cap with cassette extender) for more gravity focused 1x set ups.
Very simply, short cages look quite short and long cages look quite long, especially so when compared side by side ( indeed, your photo is of a long cage ).
Short cages typically have just a centimeter or so of space between the two pulleys, and well, you can see that a long cage has considerably more to accommodate the extra chain length needed for larger rings in either the front or back ( or both the front and back! ). If you were in a position to actually choose between buying one or the other, your LBS would surely go a long way to help, or you would have to do a considerable amount of reading: http://sheldonbrown.com/derailer-adjustment.html#chain
I do not believe there is any such thing as a "medium" cage XT derailleur, just long and extended. Medium and short cages are typically the domain of road/touring drive trains, as they use more compact gearing on the cassettes.
All modern XT derailleurs appear to have can take the same cassette range (e.g., one, two, three) while some have a little extra capacity to pick up more chain slack for multiple chainring set ups. (Among a few other subtle differences.)
I stand corrected (thank you cherouvim). What I termed long and extended appear to be officially termed "medium" and "long". A list of definitions can be found here: How to calculate the capacity of a rear derailleur. I leave the above incorrect answer intact for reference purpoes.
For the OP, here are pictures of a the Shimano XT derailleur (RD-M786) in both a medium and long version. Looking at the cage in your derailleur it is more consistent with long cage model.
Shimano XT RD-M786-GS-L (Medium)
Shimano XT RD-M786-SGS-L (Long)