Born in Ribadavia, Province of Ourense, Cortizo signed with local RC Celta de Vigo in 1956, aged 24.
[2] He only made his La Liga debut on 14 April of the following year, in a 2–1 home win against Valencia CF;[3] he featured sparingly during his spell at Balaídos, notably due to an intestine disease.
He scored his first goal in the top division on 9 December 1962, but in a 2–4 loss away to Real Madrid;[4] during his five-year tenure he won two Copa del Generalísimo trophies as well as the 1963–64 edition of the Inter-Cities Fairs Cup, being part of the Five Magnificent teams.
[5] On 27 December 1964, Cortizo broke Atlético Madrid player Enrique Collar's leg during a 3–1 victory at La Romareda.
[6] Referee Manuel Gómez Arribas did not appreciate any foul at that moment, but the player was eventually handed a 24-match ban by the Royal Spanish Football Federation, which was a national record.