  We have studied the $P\to\gamma^\star\gamma^\star$ transition form-factors
($P=\pi^0,\eta,\eta'$) within a chiral invariant framework that allows us to
relate the three form-factors and evaluate the corresponding contributions to
the muon anomalous magnetic moment $a_\mu$, through pseudoscalar pole
contributions. We use a chiral invariant Lagrangian to describe the
interactions between the pseudo-Goldstones from the spontaneous chiral symmetry
breaking and the massive meson resonances. We will consider just the lightest
vector and pseudoscalar resonance multiplets. Photon interactions and flavor
breaking effects are accounted for in this covariant framework. This article
studies the most general corrections of order $m_P^2$ within this setting.
Requiring short-distance constraints fixes most of the parameters entering the
form-factors, consistent with previous determinations. The remaining ones are
obtained from a fit of these form-factors to experimental measurements in the
space-like ($q^2\le0$) region of photon momenta. The combination of data,
chiral symmetry relations between form-factors and high-energy constraints
allows us to determine with improved precision the on-shell $P$-pole
contribution to the Hadronic Light-by-Light scattering of the muon anomalous
magnetic moment: we obtain $a_{\mu}^{P,HLbL}=(8.47\pm 0.16)\cdot10^{-10}$ for
our best fit. This result was obtained excluding BaBar $\pi^0$ data, which our
analysis finds in conflict with the remaining experimental inputs. This study
also allows us to determine the parameters describing the $\eta-\eta'$ system
in the two-mixing angle scheme and their correlations. Finally, a preliminary
rough estimate of the impact of loop corrections ($1/N_C$) and higher vector
multiplets (asym) enlarges the uncertainty up to $a_\mu^{P,HLbL} = (8.47\pm
0.16_{\rm sta}\pm0.09_{1/N_C}{}^{+0.5}_{-0.0}{}_{\rm asym})\cdot 10^{-10}$.
