  We report the transverse single-spin asymmetries of $J/\psi$ production at
forward and backward rapidity, $1.2<|y|<2.2$, as a function of $J/\psi$
transverse momentum ($p_T$) and Feynman-$x$ ($x_F$). The data analyzed were
recorded by the PHENIX experiment at the Relativistic Heavy Ion Collider in
2015 from $p$$+$$p$, $p$$+$Al, and $p$$+$Au collisions with transversely
polarized proton beams at $\sqrt{s_{_{NN}}}=200$ GeV. At this collision energy,
single-spin asymmetries for heavy-flavor particle production of $p$$+$$p$
collisions provide access to the spin-dependent gluon distribution and
higher-twist correlation functions inside the nucleon, such as the gluon
Qiu-Sterman and trigluon correlation functions. Proton+nucleus collisions offer
an excellent opportunity to study nuclear effects on the correlation functions.
The data indicate negative asymmetries at the two-standard-deviation level in
the $p$$+$Au data for $p_T<2$ GeV/$c$ at both forward and backward rapidity,
while in $p$$+$$p$ and $p$$+$Al collisions the asymmetries are consistent with
zero within the range of experimental uncertainties.
