The mixed boron-nitrogen clusters B3N3 and B4N4 have been studied using augmented coupled-cluster methods. Both molecules have monocyclic ground state structures (1A'1 and 1A(lg), respectively) with cumulenic-type equal bond lengths and alternating bond angles (BNB sharp, NBN blunt). The second most stable structures are linear BNBNBN(BN), with a 3PI ground state and alternating polyacetylenic-type bond lengths. B3N3 and B4N4 have intense infrared bands predicted to fall in the 1550-1650 and 1700-1800 cm-1 regions, respectively. Both B3N3 and B4N4 are stable, with SIGMAD(e) = 697.0 and 995.6 kcal/mol, respectively.