The energy dependence of the single-transverse-spin asymmetry, A(N), and the cross section for neutron production at very forward angles were measured in the PHENIX experiment at the Relativistic Heavy Ion Collider for polarized p + p collisions at root s = 200 GeV. The neutrons were observed in forward detectors covering an angular range of up to 2.2 mrad. We report results for neutrons with a momentum fraction of x(F) = 0.45 to 1.0. The energy dependence of the measured cross sections were consistent with x(F) scaling, compared to measurements by an experiment at the Intersecting Storage Ring, which measured neutron production in unpolarized p + p collisions at root s = 30.6-62.7 GeV. The cross sections for large x(F) neutron production for p + p collisions, as well as those in e + p collisions measured at the Hadron-Electron Ring Accelerator, are described by a pion exchange mechanism. The observed forward neutron asymmetries were large, reaching A(N) = -0.08 +/- 0.02 for x(F) = 0.8; the measured backward asymmetries, for negative x(F), were consistent with zero. The observed asymmetry for forward neutron production is discussed within the pion exchange framework, with interference between the spin-flip amplitude due to the pion exchange and nonflip amplitudes from all Reggeon exchanges. Within the pion exchange description, the measured neutron asymmetry is sensitive to the contribution of other Reggeon exchanges even for small amplitudes.