  We discuss an impact of various (path-wise) reflection-from-the barrier
scenarios upon confining properties of a paradigmatic family of symmetric
$\alpha $-stable L\'{e}vy processes, whose permanent residence in a finite
interval on a line is secured by a two-sided reflection. Depending on the
specific reflection "mechanism", the inferred jump-type processes differ in
their spectral and statistical characteristics, like e.g. relaxation
properties, and functional shapes of invariant (equilibrium, or asymptotic
near-equilibrium) probability density functions in the interval. The analysis
is carried out in conjunction with attempts to give meaning to the notion of a
reflecting L\'{e}vy process, in terms of the domain of its motion generator, to
which an invariant pdf (actually an eigenfunction) does belong.
