Underneath and integral with the conveyor (2) that feeds the rolls (R) horizontally to the machine, is a unit (20) that can be raised, into alignment with the roll feeding conveyor (1), and that turns said rolls through ninety degrees to deposit them in single file and vertically in straight horizontal channels (227) where the sets of rolls (R') are taken at the right moment by the pushing cross pieces (30) travelling along the lower side of a parallel-chain conveyor (31) driven in phase with the machine, stably mounted in the latter, and feeding the sets of rolls in phase to the subsequent means of stacking and packing. The roll orienting unit is designed to turn with variable speed and in phase with the machine cycle, to form sets made up of a selectable number of rolls. The pushing cross pieces (30) of said conveyor, by which the sets of vertical rolls are carried forward, are designed for quick removal when the machine is used for packing horizontal rolls, in order to allow for the passage through said conveyor of the conveyor (2) mounted above the orienting unit (20), which is lowered.