  In classical Bianchi-I spacetimes, underlying conditions for what dictates
the singularity structure - whether it is anisotropic shear or energy density,
can be easily determined from the generalized Friedmann equation. However, in
non-singular bouncing anisotropic models these insights are difficult to obtain
in the quantum gravity regime where the singularity is resolved at a
non-vanishing mean volume which can be large compared to the Planck volume,
depending on the initial conditions. Such non-singular models may also lack a
generalized Friedmann equation making the task even more difficult. We address
this problem in an effective spacetime description of loop quantum cosmology
(LQC) where energy density and anisotropic shear are universally bounded due to
quantum geometry effects, but a generalized Friedmann equation has been
difficult to derive due to the underlying complexity. Performing extensive
numerical simulations of effective Hamiltonian dynamics, we bring to light a
surprising, seemingly universal relationship between energy density and
anisotropic shear at the bounce in LQC. For a variety of initial conditions for
a massless scalar field, an inflationary potential, and two types of ekpyrotic
potentials we find that the values of energy density and the anisotropic shear
at the quantum bounce follow a novel parabolic relationship which reveals some
surprising results about the anisotropic nature of the bounce, such as the
maximum value of the anisotropic shear at the bounce is reached when the energy
density reaches approximately half of its maximum allowed value. The
relationship we find can prove very useful for developing our understanding of
the degree of anisotropy of the bounce, isotropization of the post-bounce
universe, and discovering the modified generalized Friedmann equation in
Bianchi-I models with quantum gravity corrections.
