  Anisotropies generically dominate the earliest stages of expansion of a
homogeneous universe. They are particularly relevant in bouncing models, since
shears grow in the contracting phase of the cosmos, making the isotropic
situation unstable. This paper extends the study of cosmological perturbations
in loop quantum cosmology (LQC) to anisotropic Bianchi I models that contain a
bounce followed by a phase of slow-roll inflation. We show that, although the
shear tensor dilutes and the universe isotropizes soon after the bounce, cosmic
perturbations retain memory of this short anisotropic phase. We develop the
formalism needed to describe perturbations in anisotropic, effective LQC, and
apply it to make predictions for the cosmic microwave background (CMB), while
respecting current observational constraints. We show that the anisotropic
bounce induces: (i) anisotropic features in all angular correlation functions
in the CMB, and in particular a quadrupolar modulation that can account for a
similar feature observed in the temperature map by the Planck satellite, and
(ii) quantum entanglement between scalar and tensor modes, that manifests
itself in temperature-polarization (T-B and E-B) correlations in the CMB.
