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Tachtsidis spent the 2013/14 Serie A season on loan at Torino but, after the Granata opted against activating a purchase option on the 23-year-old, Genoa decided to buy his full registration from Sicilian club Catania.
Free-kick specialist Lodi was only sold to Genoa by Catania last summer yet heads back to the Stadio Angelo Massimino after the co-ownership agreement between the clubs was resolved as part of the deal for Tachtsidis.
A statement published on Genoa's official website read: "Genoa announce they have consensually resolved two co-ownership agreements with Catania, acquiring the Greece international Panagiotis Tachtsidis outright with the same deal conceding our rights to retain midfielder Francesco Lodi."
Lodi, 30, will now play his club football in Serie B given Catania dropped out of the top flight in May.