Why is a contra-rotating High Pressure section more efficient?
These marketing type documents don't always mean "efficiency" in the true technical definition of the word, nor is it clear if they are talking about turbine stage efficiency (i.e. efficiency of the nozzle/turbine blade expansion process), or overall engine thermal efficiency.

The main advantage of contra-rotating is the removal of the nozzle guide vanes (NGVs) between the high pressure turbine and low pressure turbine. These LPT inlet nozzles require cooling air, that needs to be from a reasonably high pressure source because it has to be at a higher pressure than where it is being introduced back into the hot gas stream. Hence, it is usually bleed from within the high pressure compressor exit. (See reference below to "HPC S7 Air" - which means stage 7, this is the GE90 engine which has 9 or 10 HPC stages depending on the model). 

In comparision, HPT inlet nozzle guide vane cooling air has to be bleed from the HPC exit, as an even higher pressure is needed. (see reference to "HPC exit air", or "CDP", compressor discharge pressure).

(Pic from here)


Given the LPT nozzle guide vanes require cooling, removing them reduces the cooling air required, and hence this air, which has been compressed which required turbine energy to do so, and hence represents an energy investment by the engine, is still available in the combustor, and for extracting (recovering) work from in the high pressure turbine. Therefore, the thermodymanic cycle efficiency of the engine is increased, as parasitic cooling air requirements have decreased.