Monoenergetic neutron production by nuclear reactions among light elements and the production of white neutrons by such reactions are of particular interest for fusion applications. Data reduction of continuous neutron spectra is generally hampered by a lack of adequate background spectra. To find the best background spectrum for the measurement of 3H(t,n) double-differential cross sections, much effort was applied to determining a reliable background spectrum stemming from a tritium gas cell. Since the measurement of the 2H(t,n)4He reaction that was used for the efficiency determination used the same gas cell, the same background spectra could be used, and continuous neutron spectra stemming from the three-body (n+X+Y) reactions of 2H(t,n)X+Y could be extracted reliably. Thus, double-differential three-body neutron production cross sections were determined at 5.97, 7.47, 10.45, and 16.41 MeV, at angles between 0 and 90 deg with a scale uncertainty of <4%. Corresponding data with projectile and target particles exchanged are available in the same center-of-mass energy range with uncertainties of ~25%.