Mitotic exit in Saccharomyces cerevisiae and septation in Schizosaccharomyces pombe are regulated by a conserved signaling network called the mitotic exit and septum initiation networks (SIN), respectively. The network is active on one of the two anaphase B spindle-pole bodies (SPBs). Whereas the inherent asymmetry of growth by budding accounts for elements of the asymmetry in S. cerevisiae, it has been unclear how, or why, the pathway is asymmetric in S. pombe. We show that elements of SPB duplication in S. pombe are conservative, and that the SIN is active on the new SPB. SIN association with the new SPB persists after transient depolymerization of microtubules. The localization of the NIMA-related kinase, Fin1, reveals further complexity in SPB inheritance. Fin1 associates with the SPB bearing the older components in all cells and with the "new" SPB in half of the population. Fin1 only binds the new SPB when this new SPB has arisen from the duplication of an SPB that is two or more cycles old. Thus, each of the four SPBs generated over two consecutive cell cycles are different, because they have distinct fates in the next cell cycle. Fin1 binds the SPB once the SIN is active and the association requires the SIN inhibitors Byr4 and Cdc16. Fin1 physically associates with Byr4. Compromising Fin1 function leads to SIN activation on both anaphase B SPBs and promotes septation, indicating that Fin1 restrains SIN activity on the old SPB.