  Among all the PT-symmetric potentials defined on complex coordinate contours
C(s), the name "quantum toboggan" is reserved for those whose C(s) winds around
a singularity and lives on at least two different Riemann sheets. An enriched
menu of prospective phenomenological models is then obtainable via the mere
changes of variables. We pay thorough attention to the harmonic oscillator
example with a fractional screening and emphasize the role of the existence and
invariance of its quasi-exact states for different tobogganic C(s).
