A fluid dispenser head having a fluid duct (73, 61) and a dispenser orifice (83) from which the user can draw the dispensed fluid. The head including a closure mechanism for selectively closing the dispenser orifice (83). The closure mechanism has a closure member (93) that is displaceable between a closed position and an open position. The head also includes a non-rotary portion (6) that is prevented from turning relative to the dispenser member (4), and a rotary portion (7, 8) that can be turned relative to the non-rotary portion (6), and a displacement mechanism (69; 69′) that displaces the closure member (93; 93′) between the closed and open positions while the rotary portion (7, 8) is being turned relative to the non-rotary portion (6).