The first measurements of $x_F$-dependent single spin asymmetries of identified charged hadrons, $\pi^{\pm}$, $K^{\pm}$, and protons, from transversely polarized proton-proton collisions at 62.4 GeV at RHIC are presented. The measurements extend to high-$x_F$ ($|x_F|\sim 0.6$) in both the forward and backward directions.Large asymmetries are seen in the pion and kaon channels. The asymmetries in inclusive $\pi^{+}$ production, $A_N(\pi^+)$, increase with $x_F$ from 0 to $\sim$0.25 %at $x_F = 0.6$ and $A_N(\pi^{-})$ decrease from 0 to $\sim$$-$0.4. Even though $K^-$ contains no valence quarks, observed asymmetries for $K^-$ unexpectedly show positive values similar to those for $K^+$, increasing with $x_F$, whereas proton asymmetries are consistent with zero over the measured kinematic range. Comparisons of the data with predictions of QCD-based models are presented. The flavor dependent single spin asymmetry measurements of identified hadrons allow for stringent tests of theoretical models of partonic dynamics in the RHIC energy regime.