An orbital machine for bending tubes includes a fixed base frame on which is displaced in translation a guiding support cooperating with a feed-off arm driven in rotation around a horizontal axis and that forms an integral unit with a bending head having two tightening semi-rollers that are driven first according to a direction opposite in direction from the axis of the tube in a plane perpendicular to that of the bending plane XX′ of the tube, and secondly in rotation to achieve at least one bending on the tube, the base frame including a loading device allowing the tube to be bent to be held first in a horizontal position oriented in the direction of the bending head and secondly in a position that is fixed in rotation and in translation while the different bendings are achieved.