  We study supercontractivity and hypercontractivity of Markov semigroups
obtained via ground state transformation of non-local Schr\"odinger operators
based on generators of symmetric jump-paring L\'evy processes with Kato-class
confining potentials. This class of processes has the property that the
intensity of single large jumps dominates the intensity of all multiple large
jumps, and the related operators include pseudo-differential operators of
interest in mathematical physics. We refine these contractivity properties by
the concept of $L^p$-ground state domination and its asymptotic version, and
derive sharp necessary and sufficient conditions for their validity in terms of
the behaviour of the L\'evy density and the potential at infinity. As a
consequence, we obtain for a large subclass of confining potentials that, on
the one hand, supercontractivity and ultracontractivity, on the other hand,
hypercontractivity and asymptotic ultracontractivity of the transformed
semigroup are equivalent properties. This is in stark contrast to classical
Schr\"odinger operators, for which all these properties are known to be
different.
