Four different spin structures of two electrons and of two holes situated on the lowest Landau levels (LLLs) are taken into account to investigate possible bound states of the two-dimensional magnetic biexciton formed of two magnetoexcitons with opposite wave vectors and antiparallel dipole moments. The singlet and triplet states of the spins of two electrons and of two holes separately, as well as of two para- and two ortho-magnetoexcitons are considered. The general expressions describing the binding energy of the bound states and the normalization conditions characterized by the effective spin parameter eta = +/- 1, +/- 1/2 for the corresponding wave functions are derived. The most favorable of the four considered spin configurations happened to be the triplet-triplet spin structure of two electrons and of two holes. In its frame a metastable bound state with activation barrier comparable with two ionization potentials of the magnetoexciton is revealed.