  The NP constants of massless spin-0 fields propagating in Minkowski spacetime
are computed close to spatial and null infinity by means of Friedrich's
\emph{$i^0$-cylinder}. Assuming certain regularity condition on the initial
data ensuring that the field extends analytically to the critical sets, it is
shown that the NP constants at future $\mathscr{I}^{+}$ and past null infinity
$\mathscr{I}^{-}$ are independent of each other. In other words, the classical
NP constants at $\mathscr{I}^{\pm}$ stem from different parts of the initial
data given on a Cauchy hypersurface. In contrast, it is shown that, using a
slight generalisation of the classical NP constants, the associated quantities
($i^0$-cylinder NP constants) do not require the regularity condition being
satisfied and give rise to conserved quantities at $\mathscr{I}^{\pm}$ that are
determined by the same piece of initial data which, in turn, correspond to the
terms controlling the regularity of the field. Additionally, it is shown how
the conservation laws associated to the NP constants can be exploited to
construct, in flat space, heuristic asymptotic-system expansions which are
sensitive to the logarithmic terms at the critical sets.
