  In this work, we study the out-of-equilibrium dynamics of the paradigmatic
quasi-one-dimensional material (TaSe$_4$)$_2$I, that exhibits a transition into
an incommensurate CDW phase when cooled just below room temperature, namely at
T$_{\rm{CDW}} $= 263 K. We make use of both optical laser and free-electron
laser (FEL) based time-resolved spectroscopies in order to study the effect of
a selective excitation on the normal-state and on the CDW phases, by probing
the near-infrared/visible optical properties both along and perpendicularly to
the direction of the CDW, where the system is metallic and insulating,
respectively. Excitation of the core-levels by ultrashort X-ray FEL pulses at
47 eV and 119 eV induces reflectivity transients resembling those recorded when
only exciting the valence band of the compound - by near-infrared pulses at
1.55 eV - in the case of the insulating sub-system. Conversely, the metallic
sub-system displays a relaxation dynamics which depends on the energy of
photo-excitation. Moreover, excitation of the CDW amplitude mode is recorded
only for excitation at low-photon-energy. This fact suggests that the coupling
of light to ordered states of matter can predominantly be achieved when
directly injecting delocalized carriers in the valence band, rather than
localized excitations in the core levels. On a complementary side, table-top
experiments allow us to prove the quasi-unidirectional nature of the CDW phase
in (TaSe$_4$)$_2$I, whose fingerprints are detected along its $c$-axis only.
Our results provide new insights on the symmetry of the ordered phase of
(TaSe$_4$)$_2$I perturbed by a selective excitation, and suggest a novel
approach based on complementary table-top and FEL spectroscopies for the study
of complex materials.
