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YY Gem is a short-period eclipsing binary system containing two nearly identical, rapidly rotating, very active early M dwarfs. This binary represents an important benchmark system for calibrating empirical relations between fundamental properties of low-mass stars and for testing theories of interior structure and evolution of these objects. Both components of YY Gem exhibit inflated radii, which has been attributed to poorly understood magnetic activity effects. Despite a long history of magnetic activity studies of this system, no direct magnetic field measurements have been made for it. Here we present a comprehensive characterization of the surface magnetic field in both components of YY Gem. We reconstructed the global field topologies with the help of a tomographic inversion technique applied to high-resolution spectropolarimetric data. This analysis revealed moderately complex global fields with a typical strength of 200–300 G and anti-aligned dipolar components. A complementary Zeeman intensification analysis of the disentangled intensity spectra showed that the total mean field strength reaches 3.2–3.4 kG in both components of YY Gem. We used these results together with other recent magnetic field measurements of M dwarfs to investigate the relation between the global and small-scale fields in these stars. We also assessed predictions of competing magnetoconvection interior structure models developed for YY Gem, finding that only one of them anticipated the surface field strength compatible with our observations. Results of our starspot mapping of YY Gem do not support the alternative family of theoretical stellar models, which attempts to explain the radius inflation by postulating a large spot filling factor. [ABSTRACT FROM AUTHOR]