Why are contra-rotating jet engines so rare?
If the low-speed and high-speed spool rotate in the same direction, the relative speed between them is low. Contra-rotating means you need to double the individual speeds to arrive at the relative speed, and this will increase bearing losses greatly. Both friction and wear will go up with little benefit in regular operation. A notable exception is the Rolls-Royce Pegasus engine for the Harrier jump jet.

Stators between the rotating stages ensure that air flow inside the engine is mainly in axial direction, so aerodynamically a different rotation direction between two spools will make little difference. However, contra-rotation will indeed make the stator between the last stage of the high- and the first stage of the low-pressure turbine obsolete. The video linked by @Moo cites a 10% reduction in part count as the advantage. Additionally, the first stage of the low-pressure turbine should show improved efficiency (like the second propeller of a contra-rotating propeller).