  We study the decays of the pseudotensor mesons $[ \pi_{2}(1670) , K_{2}(1770)
, \eta_{2}(1645) , \eta_{2}(1870) ]$ interpreted as the ground-state nonet of
$1^1 D_{2}$ $\bar{q}q$ states using interaction Lagrangians which couple them
to pseudoscalar, vector, and tensor mesons. While the decays of $\pi_2 (1670)$
and $K_2 (1770)$ can be well described, the decays of the isoscalar states
$\eta_2 (1645)$ and $\eta_2 (1870)$ can be brought in agreement with
experimental data only if the mixing angle between nonstrange and strange
states is surprisingly large (about $-42^\circ$, similar to the mixing in the
pseudoscalar sector, in which the chiral anomaly is active). Such a large
mixing angle is however at odd with all other conventional quark-antiquark
nonets: if confirmed, a deeper study of its origin will be needed in the
future. Moreover, the $\bar{q}q$ assignment of pseudotensor states predicts
that the ratio $[ \eta_2 (1870) \rightarrow a_2 (1320) \pi]/[\eta_2 (1870)
\rightarrow f_2 (1270) \eta]$ is about $23.5$. This value is in agreement with
Barberis et al., ($20.4 \pm 6.6$), but disagrees with the recent reanalysis of
Anisovich et al., ($1.7 \pm 0.4$). Future experimental studies are necessary to
understand this puzzle. If Anisovich's value shall be confirmed, a simple nonet
of pseudoscalar mesons cannot be able to describe data (different assignments
and/or additional state, such as an hybrid state, will be needed). In the end,
we also evaluate the decays of a pseudoscalar glueball into the aforementioned
conventional $\bar{q}q$ states: a sizable decay into $K^\ast_2 (1430) K$ and
$a_2 (1230) \pi$ together with a vanishing decay into pseudoscalar-vector pairs
[such as $\rho(770) \pi$ and $K^\ast (892) K$] are expected. This information
can be helpful in future studies of glueballs at the ongoing BESIII and at the
future PANDA experiments.
