The machine (1; 100) comprises a winding station (3; 101), adapted to receive secondary winding cores (T) coaxial with, and adjacent to, one another. A cutting device (11; 104) is also provided, with a plurality of blades (13; 105), arranged upstream of the winding station (3; 101) with respect to the feeding direction of the web material (N) and adapted to subdivide the web material into plurality of strips (S1-S5) of web material; The machine has a web material evaluation system and a programmable unit (71) for processing data collected by the web material evaluation system.