Background: The possible existence of a phase transition to a ferromagnetic state in neutron matter as origin of the extremely high magnetic fields of neutron stars is still an open issue. Whereas many phenomenological interactions predict this transition at densities accessible in neutron stars, microscopic calculations based on realistic interactions show no indication of it. The existence or non-existence of this transition is a consequence of the different role of nucleon-nucleon correlations in polarized and unpolarized neutron matter. Therefore, to give a definite answer to this issue it is necessary to analyze the behavior of these correlations. Purpose: Using the Hellmann-Feynman theorem we analyze the contribution of the different terms of the nucleon-nucleon interaction to the spin symmetry energy of neutron matter with the purpose of identifying the nature and role of correlations in polarized and unpolarized neutron matter. Methods: The analysis is performed within the microscopic Brueckner-Hartree-Fock approach using the Argonne V18 realistic potential plus the Urbana IX three-body force. Results: Our results show no indication of a ferromagnetic transition as the spin symmetry energy of neutron matter is always an increasing function of density. They show also that the main contribution to it comes from the S = 0 channel, acting only in non-polarized neutron matter, in particular from the S-1(0) and the D-1(2) partial waves. Three-body forces are found to play a secondary role in the determination of the spin symmetry energy. Conclusions: By evaluating the kinetic energy difference between the correlated system and the underlying Fermi sea to estimate the importance of correlations in spin-polarized neutron matter, we conclude that non-polarized neutron matter is more correlated than totally polarized one.