Recently published measurements from the PHENIX experiment at the Relativistic Heavy Ion Collider will be presented for the transverse single-spin asymmetries (TSSAs) of direct photons, neutral pions, and eta mesons produced at midrapidity in 200 GeV proton-proton collisions. As hadrons, neutral pions and eta mesons are sensitive to nonperturbative spin-momentum correlations both within the proton and in the process of hadronization; at midrapidity they probe the dynamics of gluons along with a mix of quark flavors. The direct photon TSSA measurement instead isolates initial-state spin-momentum correlations within the proton and at midrapidity provides a clean probe of the gluon dynamics in transversely polarized protons. All three of these results will help to constrain the collinear twist-3 trigluon correlation function. The direct photon result is the first-ever measurement of this observable from a hadron collider, and all three of the measurements represent the final, definitive results for these observables from the PHENIX experiment, which completed data taking with polarized beams in 2015.