The premise of the question is incorrect. There has never been an official flag of Mexico featuring an eagle facing forward (toward the viewer).
Since the first national flag (1821), the Mexican coat of arms has depicted an eagle in profile, facing the viewer's right (heraldic left / dexter). It stands on a nopal cactus growing from a rock in a lake, devouring a serpent—symbolizing the Aztec founding myth of Tenochtitlan.
In every version, the eagle is shown in left profile (head turned to its right). A front-facing eagle would violate Mexican heraldic tradition and the legal specifications (Ley sobre el Escudo, la Bandera y el Himno Nacionales).