Periodoannulene molecules and ions CxIxq in planar geometry offer examples of systems with the potential for outer σ and inner πring-current double aromaticity, given a sufficient overlap of tangential pσ-orbital manifolds on the large atoms of the outer cycle. Previous theoretical work indicated concentric diatropic currents in the dication C6I62+. Ab initio ipsocentric calculations support an account in terms of frontier-orbital selection rules for current contributions in C6I62+ (and radical C6I6+, implicated in recent experimental work on the oxidation of periodobenzene). A σ/πanalogue of the annulene-within-an-annulene model is applied here to periodo systems based on cyclooctatetraene. Model species C8I8q with charges q = 0, +1, +2, +4, -2 and structures constrained to a planar D4h symmetry exhibit maps with all combinations of σ/πcon- and counter-rotation, comprising global σ ring currents on the iodine perimeter and central πring currents on the carbocycle. All can be rationalized by the separate application of the tropicity selection rules to the two subsystems, whether in singlet or triplet states.