An annular thread cutting die head (26) for a power driven threading machine is comprised of an annular support plate (32) supporting a plurality of radially displaceable thread cutting dies (44) spaced apart about the axis (A) of the die hed (26), and an annular cam plate assembly (34) coaxial with and pivotal relative to the support plate (32) to radially displace the cutting dies (44). The cam plate assembly (34) is spring biased to pivot relative to the support plate (32) in the direction to radially retract the cutting dies (44), and is selectively operable relative to the support plate (32) to achieve the cutting of a straight thread or to achieve the cutting of a tapered thread. In connection with the cutting of a tapered thread, receding replacement of the cutting dies (44) is controlled by a unique cam arrangement (132) on the machine carriage (10) engaged by the cam plate assembly (34) under the spring bias and independent of torque imposed on the die head (26) during the threading operation.