  Understanding the spin structure of the proton is of large interest to the
nuclear physics community and it is one of the main goals of the spin physics
program at the Relativistic Heavy Ion Collider (RHIC). Measurements from data
taken by the PHENIX detector with transverse ($p^{\uparrow} + p$,
$p^{\uparrow}$ + Al, $p^{\uparrow}$ + Au) proton polarization play an important
role in this, in particular, due to the leading order access to gluons in
polarized protons. Transverse single-spin asymmetries (TSSAs) provide insight
into initial and final state spin-momentum and spin-spin parton-hadron
correlations. In addition to possible final state contributions, $\pi^{0}$ and
$\eta$ TSSAs access both quark and gluon correlations in the polarized proton.
Furthermore, the $p^{\uparrow} + A$ data from RHIC provides an opportunity to
study the effect of TSSAs in the presence of additional nuclear matter.
Midrapidity $\pi^{0}$ and $\eta$ mesons are measured at PHENIX by detecting the
2$\gamma$ decay with the electromagnetic calorimeter (EMCal) in the central arm
spectrometer, which has fine granularity for the resolution of separate decay
photons. New results for TSSAs of midrapidity $\pi^{0}$ and $\eta$ mesons in
$\sqrt{s_{NN}} = 200$ GeV $p^{\uparrow}$ + Au and $p^{\uparrow}$ + Al
collisions from the 2015 running year will be presented, and compared with the
recent $\sqrt{s} = 200$ GeV $p^{\uparrow} + p$ results.
