We have analyzed spectra recorded between 50 and 650 cm-1 by the Composite Infrared Spectrometer (CIRS) aboard the Cassini spacecraft at low and high emission angles to determine simultaneously the H2 mole fraction and ortho-to-para ratio in Titan's troposphere. We used constraints from limb spectra between 50 and 900 cm-1 and from in situ measurements by the Huygens probe to characterize the temperature, haze and gaseous absorber profiles. We confirm that the N2-CH4 collision-induced absorption (CIA) coefficients used up to now need to be increased by about 52% at temperatures of 70-85 K. We find that the N2-N2 CIA coefficients are also too low in the N2 band far wing, beyond 110 cm-1, in agreement with recent quantum mechanical calculations. We derived a H2 mole fraction equal to (0.88 ± 0.13) × 10-3, which pertains to the ~1-34 km altitude range probed by the S0(0) and S0(1) lines. This result agrees with a previous determination based only on the H2-N2 dimer transition in the S0(0) line, and with the in situ measurement by the Gas Chromatograph Mass Spectrometer (GCMS) aboard Huygens. It is 3-4 times smaller than the value measured in situ by the Ion Neutral Mass Spectrometer (INMS) of Cassini at 1000-1100 km. The H2 para fraction is close to equilibrium in the 20-km region. CIRS spectra can be fitted assuming ortho-to-para (o-p) H2 thermodynamical equilibrium at all levels or a constant para fraction in the range 0.49-0.53. We have investigated different mechanisms that may operate in Titan's atmosphere to equilibrate the H2 o-p ratio and we have developed a one-dimensional model that solves the continuity equation in presence of such conversion mechanisms. We conclude that exchange with H atoms in the gas phase or magnetic interaction of H2 in a physisorbed state on the surface of aerosols are too slow compared with atmospheric mixing to play a significant role. On the other hand, magnetic interaction of H2 with CH4, and to a lesser extent N2, can operate on a timescale similar to the vertical mixing time in the troposphere. This process is thus likely responsible for the o-p equilibration of H2 in the mid-troposphere implied by CIRS measurements. The model can reproduce the inferred o-p ratio in the 20-km region, assuming low atmospheric mixing in the troposphere down to 15-20 km and conversion rates with CH4 or N2 slightly larger than obtained from an extrapolation of natural ortho-para conversion rate measured in gaseous hydrogen.