Epitaxial thin films of Bi2FeCrO6 (BFCO) have been synthesized by pulsed laser deposition on SrRuO3 on (100)- and (111)-oriented SrTiO3 substrates. Detailed x-ray diffraction and cross-section transmission electron microscopy analysis revealed a double perovskite crystal structure of the BFCO epitaxial films very similar to that of BiFeO3 along with a particularly noteworthy Fe3+/Cr3+ cation ordering along the  direction. The films contain no detectable magnetic iron oxide impurities and have the correct cationic average stoichiometry throughout their thickness. They however exhibit a slight modulation in the Fe and Cr compositions forming complementary stripe patterns, suggesting minor local excess or depletion of Fe and Cr. The epitaxial BFCO films exhibit good ferroelectric and piezoelectric properties, in addition to magnetic properties at room temperature, as well as an unexpected crystallographic orientation dependence of their room-temperature magnetic properties. Our results qualitatively confirm the predictions made using the ab initio calculations: the double perovskite structure of BFCO films exhibit a Fe3+/Cr3+ cation ordering and good multiferroic properties, along with the unpredicted existence of magnetic ordering at room temperature.