An apparatus and method for refinishing a tapered surface of a machine tool component. The apparatus includes a head defining at least three tapered segments which are configured to conform precisely to the taper of the machine tool component being refinished. Each segment includes a surface for receiving an abrasive strip, the surfaces configured to conform and confront a conical surface of a taper to be refinished. The head is adapted to engage the tapered surface of the machine tool component so that upon rotation of the head, the abrasive strips abrade the conical surface of the component. A head for refinishing a spindle bore includes three outwardly extending segments whereas the head for refinishing a tool holder shank surface includes three segments extending between annular portions, each segment having an inner tapered surface for receiving the abrasive strips.