  We present a detailed study of of a global bifurcation occuring in a
turbulent von K\'arm\'an swirling flow. In this system, the statistically
steady states progressively display hysteretic behaviour when the Reynolds
number is increased above the transition to turbulence. We examine in detail
this hysteresis using asymmetric forcing conditions --- rotating the impellers
at different speeds. For very high Reynolds numbers, we study the sensitivity
of the hysteresis cycle --- using complementary Particle Image Velocimetry
(PIV) and global mechanical measurements --- to the forcing nature, imposing
either the torque or the speed of the impellers. New mean states, displaying
multiple quasi-steady states and negative differential responses, are
experimentally observed in torque control. A simple analogy with electrical
circuits is performed to understand the link between multi-stability and
negative responses. The system is compared to other, similar "bulk" systems, to
understand some relevant ingredients of negative differential responses, and
studied in the framework of thermodynamics of long-range interacting systems.
The experimental results are eventually compared to the related problem of
Rayleigh-B\'enard turbulence.
