Why does the LP compressor rotate in the opposite direction of the HP compressor in some multi-spool turbine engines?
Contra-rotating spools are a very recent development, apart from the Bristol-Siddeley (later: Rolls-Royce) Pegasus engine for the Harrier. The Rolls-Royce Trent 900 started the trend when its high pressure spool was made to rotate in the opposite direction to the other two spools. From Wikipedia:
It is also the first member of the Trent family to feature a contra-rotating HP spool
Next was General Electric with the GEnx, a two-spool design with a high pressure compressor rotating clockwise while the low pressure compressor rotates counterclockwise.
Contra-rotation means that the relative speeds between the spools are increased, increasing the bearing losses between them. However, contra-rotation helps to increase efficiency - now the stator between the last low pressure stage and the first high pressure stage of the compressor and the stator between the last high pressure turbine stage and the first low pressure turbine stage can be left away which reduces parts count, engine length, engine mass and friction.
Counter-rotating spools for the reaction turbines to reduce load on guide vanes.