Twin domains are often found as structural defects in symmetry mismatched epitaxial thin films. The delafossite ABO2, which has a rhombohedral structure, is a good example that often forms twin domains. Although bulk metallic delafossites are known to be the most conducting oxides, high conductivity is yet to be realized in thin film forms. Suppressed conductivity found in thin films is mainly caused by the formation of twin domains, and their boundaries can be a source of scattering centers for charge carriers. To overcome this challenge, the underlying mechanism for their formation must be understood so that such defects can be controlled and eliminated. Here, we report the origin of structural twins formed in a CuCrO2 delafossite thin film on a substrate with hexagonal or triangular symmetries. A robust heteroepitaxial relationship is found for the delafossite film with the substrate, and the surface termination turns out to be critical to determine and control the domain structure of epitaxial delafossites. Based on such discoveries, we also demonstrate twin-free epitaxial thin films grown on high-miscut substrates. This finding provides an important synthesis strategy for growing single-domain delafossite thin films and can be applied to other delafossites for the epitaxial synthesis of high-quality thin films.