We conduct a statistical analysis of 2,094 reconnection events in Saturn's near-equatorial magnetotail previously identified in Cassini magnetometer data from intervals during 2006 and 2009/2010. These consist of tailward propagating plasmoids and planetward propagating dipolarizations, with approximately twice as many plasmoids as dipolarizations. We organize these by three related planetary period oscillation (PPO) phase systems, the northern and southern PPO phases relative to noon, the same phases retarded by a radial propagation delay, and the local retarded phases that take account of the azimuth (local time) of the observation. Clear PPO modulation is found for both plasmoid and dipolarization events, with local retarded phases best organizing the event data with the modulation in event frequency propagating across the tail as the PPO systems rotate. This indicates that the events are localized in azimuth, rather than simultaneously affecting much of the tail width. Overall, events occur preferentially by factors of ~3 at northern and southern phases where the tail current sheet is expected locally to be thinnest in the PPO cycle, with field lines contracting back from their maximum radial displacement, compared with the antiphase conditions. Separating the events into those representing the start of independent reconnection episodes, occurring at least 3 hr after the last, and events in subsequent clusters, shows that the above phases are predominantly characteristic of the majority cluster events. The phases at the start of independent reconnection episodes are typically ~60° earlier.