The final analysis of the experiment determining both components of the transverse polarization of electrons (sigma(T1), sigma(T2)) emitted in the beta decay of polarized, free neutrons is presented. The T-odd, P-odd correlation coefficient quantifying sigma(T2), perpendicular to the neutron polarization and electron momentum, was found to be R = 0.004 +/- 0.012 +/- 0.005. This value is consistent with time-reversal invariance and significantly improves both the earlier result and limits the relative strength of the imaginary scalar couplings in the weak interaction. The value obtained for the correlation coefficient associated with sigma(T1), N = 0.067 +/- 0.011 +/- 0.004, agrees with the standard model expectation, providing an important sensitivity test of the experimental setup. The present result sets constraints on the imaginary part of scalar and tensor couplings in weak interactions. Implications for parameters of the leptoquark exchange model and the minimal supersymmetric model with R-parity violation are discussed.