The machine includes: a bending head movable with at least two degrees of freedom in a transverse vertical plane; a first drive is arranged to move the bending head in the plane and to move a tool of the bending head between an open position, in which it defines with a die of the bending head a free space adapted to receive a tube to be bent, and a closed position, in which it clamps the tube against the die. A tube-carrying structure is rotatable about an axis of rotation perpendicular to the plane; a second drive is arranged to cause the tube-carrying structure to rotate about its own axis of rotation; and an electronic control unit is arranged to control the first and second drives so as to move the bending head and the tube-carrying structure in a position such that the tube is received with a desired orientation in the free space between the die and the tool.