Machine for bending a piece having longitudinal form such as a tube or a section bar, e.g. with circular, rectangular, oval section or the like, comprising a base with an upper surface, as well as power means, for developing the power and the mechanical torque necessary for bending said piece to be bent, such as a gear motor, or the like, in which said power means transmit the rotary motion to a matrix (10p; 10g) mounted on the upper surface and having substantially circular form with an annular groove whose profile substantially copies the form of the transverse half section of the piece, the machine further comprising a counter-matrix (16p, 17p; 16g, 17g) cooperating with said matrix (10p; 10g) and which is mounted in front of the matrix itself above the upper surface, in a manner so as to make the piece pass between the matrix and the counter-matrix during operation, the matrix (10p; 10g) further comprising a drive bracket (19p; 19g) and the piece being inserted during the operation of the machine also between the drive bracket (19p; 19g) and the wall of the annular groove of the matrix (10p; 10g), characterized in that the machine provides for a member which is mounted in an unthreadable or extractible manner from a vertical axle as well as rotatable around the same vertical axle above the upper surface, said member having a first side and a second side, as well as a first end and a second end, said first end of the member supporting a first counter-matrix (16p, 17p) for bending small pieces, with smaller transverse section, and said second end of the member supporting a second counter-matrix (16g, 17g) for bending large pieces, with greater transverse section, said first end and said second end of the member being substantially aligned with respect to the center of the matrix (10p; lOg) as well as with respect to said vertical shaft or axle during operation, and said member being rotatable around the vertical axle in order to selectively bring one or the other of said counter-matrices (16p, 17p or 16g, 17g) in a position such to cooperate with the respective matrix (lOp or 10g) during the operation of the machine, and in that during operation, the member assumes one of the four following positions and cannot rotate by virtue of anti-rotation means: - a first position, with the first side of the member directed upward, in which the first counter-matrix (16p, 17p) cooperates with a first matrix, for small pieces, bending a small piece to the right; - a second position, rotated 180° with respect to the first position around the vertical axle and in which the second counter-matrix (16g, 17g) cooperates with a second matrix, for large pieces, bending a large piece to the left; - a third position, with the first side of the member directed downward, in which the second counter-matrix (16g, 17g) cooperates with said second matrix, for large pieces, bending a large piece to the right; and - a fourth position, rotated 180° with respect to the third position around the vertical axle and in which the first counter-matrix (16p, 17p) cooperates with said first matrix, for small pieces, bending a small piece to the left. - a first position, with the first side of the member directed upward, in which the first counter-matrix (16p, 17p) cooperates with a first matrix, for small pieces, bending a small piece to the right; - a second position, rotated 180° with respect to the first position around the vertical axle and in which the second counter-matrix (16g, 17g) cooperates with a second matrix, for large pieces, bending a large piece to the left; - a third position, with the first side of the member directed downward, in which the second counter-matrix (16g, 17g) cooperates with said second matrix, for large pieces, bending a large piece to the right; and - a fourth position, rotated 180° with respect to the third position around the vertical axle and in which the first counter-matrix (16p, 17p) cooperates with said first matrix, for small pieces, bending a small piece to the left.