  The aim of this work is to shed light on some lesser known aspects of
Polyakov-loop--extended chiral models (namely the Polyakov-loop extended
Nambu--Jona-Lasinio and Quark-Meson models), especially on the correlation of
the quark sector with the Polyakov loop. We show that the ordering of chiral
and Polyakov-loop transitions and their difference in temperature as seen in
lattice QCD calculations could be realized with a critical scale of the
Polyakov-loop potential that is larger than the one in pure gauge theory. The
comparison of the results for the Polyakov-loop susceptibility obtained using
the self-consistent medium-dependent quark mass with those obtained while
keeping these masses at a fixed value allows to disentangle chiral-symmetry
restoration and center-symmetry breaking effects. Furthermore, a confined
chirally restored phase is identified by a plateau in the quark contribution to
thermodynamics and by sigma and pion spectral functions that coincide but have
a small width. We also discuss that, for some large chemical potential values,
the explicit center-symmetry breaking is so strong that statistical
deconfinement is realized at infinitely small temperatures. Both the missing
sensitivity of the Polyakov loop to the quark mass, except at close to the
chiral transition, and the Polyakov loop being zero at zero temperature at all
chemical potentials, can be interpreted as indications of a missing mechanism
which accounts for the quark back-reaction on the Polyakov loop.
