Classic formulations of aromaticity have long been associated with topologically planar conjugated macrocyclic systems. The theoretical possibility of so-called bicycloaromaticity was noted early on. However, it has yet to be demonstrated by experiment in a simple synthetic organic molecule. Conjugated organic systems are attractive for studying the effect of structure on electronic features. This is because, in principle, they can be modified readily through dedicated synthesis. As such, they can provide useful frameworks for testing by experiment with fundamental insights provided by theory. Here we detail the synthesis and characterization of two purely organic non-planar dithienothiophene-bridged [34]octaphyrins that permit access to two different aromatic forms as a function of the oxidation state. In their neutral forms, these congeneric systems contain competing 26 and 34 π-electronic circuits. When subject to two-electron oxidation, electronically mixed [4n+1]/[4n+1] triplet biradical species in the ground state are obtained that display global aromaticity in accord with Baird's rule.