A measurement of the 2H(d,2He)2n charge-exchange reaction at Ed=171 MeV has been performed in order to study the interaction of the di-neutron system. Spectroscopy of the neutron-neutron final-state interaction with a resolution of about 115 keV (FWHM) was achieved by measuring the momentum of 2He at forward scattering angles. In the experiment the two protons of the unbound 2He are momentum analyzed with a magnetic spectrometer and detected with the same detector. A simple reaction theory is employed in order to obtain information about the neutron-neutron interaction from the measured cross section data. The validity of the impulse approximation for the d?2He transition at intermediate energies is discussed and compared to the Watson-Migdal theory of the proton-proton final-state interaction. Predictions of the impulse approximation for the analog transition d?2n are confronted with experimental data. A procedure to extract the neutron-neutron scattering length ann from this comparison is presented. The result is compatible with the recommended value of ann=-18.6(�0.4) fm.