The cyclodimerization of p-toluenesulfonamide and 3-chloro-2-(chloromethyl)-1-propene to prepare N,N'-bis(p-toluenesulfonyl)-3,7-bis(methylene)-1,5-diazacyclooctane (1a) and its ozonation to the corresponding 3,7-dione 2a are reported. Unusual transannular cyclizations initiated by lithium aluminum hydride treatment or bromination of 1a and oxidative coupling of the dioxime derived from 2a are described. These reactions lead, respectively, to the following derivatives of the little-studied 3,7-diazabicyclo[3.3.0]octane ring system: 1,5-dimethyl-3,7-diazabicyclo[3.3.0]octane (5), N,N'-bis(p-toluenesulfonyl)-1,5-bis(bromomethyl)-3,7-diazabicyclo[3.3.0]octane (8), and N,N'-bis(p-toluenesulfonyl)-1,5-dinitro-3,7-diazabicyclo[3.3.0]octane, (12). Acid-catalyzed hydration of 1a, in contrast, gives the expected 5-methyl-3,7-diazabicyclo[3.3.1]nonan-1-ol (10). Reaction of the dibromide 8 with the nucleophiles, sodium sulfide, sodium oxide, and sodium p-toluenesulfonamide conveniently delivers the corresponding novel 3,7,10-triheterocyclic [3.3.3]propellanes.