  The strictly classical propagation of an initial Wigner function, referred to
as TWA or LSC-IVR, is considered to provide approximate averages, despite not
being a true Wigner function: it does not represent a positive operator. We
here show that its symplectic Fourier transform, the truncated chord
approximation (TCA), coincides with the full semiclassical approximation to the
evolved quantum characteristic function (or chord function) in a narrow
neighbourhood of the origin of the dual chord phase space. Surprisingly, this
small region accounts for purely quantum features, such as blind spots and
local wave function correlations, as well as the expectation of observables
with a close classical correspondence. Direct numerical comparison of the TCA
with exact quantum results verifies the semiclassical predictions for an
initial coherent state evolving under the Kerr Hamiltonian. The resulting clear
criterion for any further features, which may be estimated by classical
propagation, is that, within the chord representation, they are concentrated
near the origin.
