The nearby M dwarf binary GJ65 AB, also known as BL Cet and UV Cet, is a unique benchmark for investigation of dynamo-driven activity of low-mass stars. Magnetic activity of GJ65 was repeatedly assessed by indirect means, such as studies of flares, photometric variability, X-ray, and radio emission. Here, we present a direct analysis of large-scale and local surface magnetic fields in both components. Interpreting high-resolution circular polarization spectra (sensitive to a large-scale field geometry) we uncovered a remarkable difference of the global stellar field topologies. Despite nearly identical masses and rotation rates, the secondary exhibits an axisymmetric, dipolar-like global field with an average strength of 1.3 kG while the primary has a much weaker, more complex, and non-axisymmetric 0.3 kG field. On the other hand, an analysis of the differential Zeeman intensification (sensitive to the total magnetic flux) shows the two stars having similar magnetic fluxes of 5.2 and 6.7 kG for GJ65 A and B, respectively, although there is evidence that the field strength distribution in GJ65 B is shifted toward a higher field strength compared to GJ65 A. Based on these complementary magnetic field diagnostic results, we suggest that the dissimilar radio and X-ray variability of GJ65 A and B is linked to their different global magnetic field topologies. However, this difference appears to be restricted to the upper atmospheric layers but does not encompass the bulk of the stars and has no influence on the fundamental stellar properties.