  Axions currently provide the most compelling solution to the strong CP
problem. These particles may be copiously produced in the early universe,
including via thermal processes. Therefore, relic axions constitute a hot dark
matter component and their masses are strongly degenerate with those of the
three active neutrinos, as they leave identical signatures in the different
cosmological observables. In addition, thermal axions, while still relativistic
states, also contribute to the relativistic degrees of freedom, parameterised
via $N_{eff}$. We present the cosmological bounds on the relic axion and
neutrino masses, exploiting the full Planck mission data, which include
polarization measurements. In the mixed hot dark matter scenario explored here,
we find the tightest and more robust constraint to date on the sum of the three
active neutrino masses, $\sum m_\nu <0.136$ eV at $95\%$ CL, obtained in the
well-known linear perturbation regime. The Planck Sunyaev-Zeldovich cluster
number count data further tightens this bound, providing a $95\%$ CL upper
limit of $\sum m_\nu <0.126$ eV in this very same mixed hot dark matter model,
a value which is very close to the expectations in the inverted hierarchical
neutrino mass scenario. Using this same combination of data sets we find the
most stringent bound to date on the thermal axion mass, $m_a<0.529$ eV at
$95\%$ CL.
