<p>The roller (8) has an axial section comprising a folding surface to fold a flat product i.e. signature (S). Another axial section has a folding surface to fold the signature and a main clearance (44) radially recessed with respect to the folding surfaces. The former axial section has an auxiliary clearance radially recessed with respect to the former folding surface. The main clearance overlaps a lateral part of a front edge of the signature when the signature is received by the roller.</p>