For the two reactions H- + D-2 --> D- + HD and D- + H-2 --> H-+ HD, absolute integral and differential cross sections have been determined in a guided ion beam (GIB) apparatus. From the effective cross sections, measured in the translational energy range E-T = 0.1-10 eV, thresholds for the onset of the reactions have been derived, indicating effective barrier heights of 350 +/- 60 meV for H- + D-2 and 330 +/- 60 meV for D- + H-2. These values are substantially lower than previously reported experimental and theoretical data. Comparison of the integral cross sections for the two isotopic variants reveals a significant isotope effect. The collision of the light ion with the heavy molecule leads to a cross section which is almost a factor of 2 smaller than for D- reacting with H-2. Possible experimental effects discriminating D- versus H- are discussed. The recently reported differential cross section for H- + D-2 which has been obtained with a crossed beam experiment is in good accordance with the present data determined with the GIB technique. A new result is that, at small collision energies, the angular dependence is very similar for both isotopic variants. In view of the quite large isotopic effect for the integral cross sections this is not self-evident. All these detailed results pose a challenge to full quantum dynamical scattering theory of the H-3(-) collision system.