A grooving or part-off tool assembly for Swiss-style machining includes a holder. The holder includes a resilient normally-closed holder clamp. In a preferred embodiment the holder clamp biases a blade holding a cutting insert against a pocket side abutment surface of a holder pocket.

A cutting tool assembly having a longitudinal axis establishing upward and downward directions, includes a Y-axis parting adaptor mounted onto a holder. The parting adaptor has an insert pocket including a base jaw configured to support a cutting insert. The holder has a holder head portion and a shank portion. The holder head portion has an adaptor pocket in which the parting adaptor is mounted. If the holder shank cross-sectional shape is square or rectangular the forwardmost cutting edge is directly above the forwardmost holder shank surface, and if the holder shank cross-sectional shape is round and a centrally extending holder shank axis extends to the forwardmost cutting edge.