  We study the contraction properties (up to shift) for admissible
Rankine-Hugoniot discontinuities of $n\times n$ systems of conservation laws
endowed with a convex entropy. We first generalize the criterion developed in
[47], using the spatially inhomogeneous pseudo-distance introduced in [50]. Our
generalized criterion guarantees the contraction property for extremal shocks
of a large class of systems, including the Euler system. Moreover, we introduce
necessary conditions for contraction, specifically targeted for intermediate
shocks. As an application, we show that intermediate shocks of the
two-dimensional isentropic magnetohydrodynamics do not verify any of our
contraction properties. We also investigate the contraction properties, for
contact discontinuities of the Euler system, for a certain range of contraction
weights.
  All results do not involve any smallness condition on the initial
perturbation, nor on the size of the shock.
