A reclosable opening device (3, 3') for a sealed package (1) of a pourable food product, which opening device has a frame (10) fitted about a pierceable portion (4) of the package (1) and defining a through pour opening (11); a removable threaded cap (12) that screws onto the frame (10) to close the pour opening (11); a tubular cutter (15) engaging the pour opening (11) and having, at one axial end, cutting means (31) which cooperate with the pierceable portion (4) to unseal the package (1); first connecting means (13) connecting the cap (12) to the cutter (15), and which, as the cap (12) is unscrewed off the frame (10), push the cutter (15) towards the pierceable portion (4); and second connecting means (14) connecting the frame (10) to the cutter (15), and which, in use, feed the cutter(15) along a predetermined piercing path (P) through the pierceable portion (4) in response to unscrewing of the cap (12); and the piercing path (P) of the cutter (15), as the cap (12) is unscrewed off the frame (10), has a first portion (P1) of pure axial translation, followed by a second portion (P2) having both an axial and a rotary component of motion.