A recently published sample of 21 detached eclipsing binaries in the Small Magellanic Cloud provides a valuable test of the binary mass function for massive stars. We show that 50% of detached binaries have companions with very similar masses, q = M2/M1 > 0.87, where M1 and M2 denote the masses of the two binary components, M1 ≥ M2. A Salpeter relative mass function for the secondary is very strongly excluded, and the data are consistent with a flat mass function containing 55% of the systems and a "twin" population with q > 0.95 containing the remainder. We survey the existing literature on binary mass ratios and conclude that a significant twin population (of order 20%-25%) exists in binaries that are likely to interact across a broad range of stellar masses and metallicity. Interactions involving twins have distinctly different properties from those involving stars of unequal mass; the secondaries will tend to be evolved, and common-envelope evolution is qualitatively different. The implications of such a population for both binary interactions and star formation are substantial, and we present some examples. We argue that twin systems may provide a natural stellar population to explain the recently proposed prompt channel for Type Ia supernovae, and the presence of a twin population dramatically reduces the maximum inferred merger rate between neutron stars (NSs) and black holes relative to the NS-NS merger rate. Twins may also be important for understanding the tendency of white dwarf and NS binaries to be nearly equal in mass, and inclusion of twins in population studies will boost the blue straggler production rate.