In a tool-holding turret for a rotary machine, such as a lathe, having a base member disposed on a cross-slide rest and rotatable in relation thereto about a first axis of rotation, a rotary turret head being disposed on said base member, and rotating about a second axis of rotation, inclined to said first axis of rotation at an offset angle, the improvement wherein said turret head has a first ring of tools and a second ring of tools, each said ring having the shank axis of its tools lying on the conical surface of an imaginary cone, there being a separate cone for each said ring, the cones being mirror images of each other;
- the vertices of the two cones coinciding and both lying on the said second axis of rotation, the sum of half the vertex angle of the cone plus the offset angle of said second axis from said first axis being a right angle.