  We analyze the Bianchi I cosmology in the presence of a massless scalar field
and describe its dynamics via a semiclassical and quantum polymer approach. We
study the morphology of the Big Bounce by adopting three different sets of
configurational variables: the Ashtekar connections, a set of anisotropic
volume-like coordinates and the Universe volume plus two anisotropy coordinates
(the latter two sets of variables would coincide in the case of an isotropic
Universe). In the semiclassical analysis we demonstrate that the value of the
critical matter energy density depends on the Cauchy problem for the dynamics
when adopting the Ashtekar connections or the anisotropic volume-like
coordinates. On the contrary, when the Universe volume is considered as a
configurational coordinate, we are able to derive a polymer-modified Friedmann
equation for the Bianchi I model, from which the expression of the critical
energy density can be derived. This analysis shows that the Big Bounce has
universal features only when the Universe volume is defined on the polymer
lattice. Then, a cosmological constant is included in the Ashtekar connections'
formulation and some interesting results are mentioned making a comparison
between the synchronous dynamics and that one when the scalar field is taken as
a relational time. From a pure quantum point of view, we investigate the
Bianchi I dynamics in terms of the Ashtekar connections. We apply the ADM
reduction of the variational principle and then we quantize the system. We
study the resulting Schr\"{o}dinger dynamics, stressing that the behavior of
the wave packet peak over time singles out common features with the
semiclassical trajectories, confirming the non-universal character of the
emerging Big Bounce also on a quantum level.
