Conformal equivalence is the relation between two Riemann surfaces that admit a biholomorphic map between them. Such a map is a bijection and is holomorphic together with its inverse function (Kovalev 2007, p. 17). It preserves the complex structure and its orientation.
Conformal equivalence is a global requirement. A conformal mapping that preserves angles locally need not be a bijection onto the specified target Riemann surface. In this usage, orientation-reversing maps do not establish conformal equivalence.
