  If two species of ultracold atoms are loaded in a sufficiently tight optical
lattice at a commensurate total filling factor, the net number-of-atoms
transport is suppressed by the Mott-Hubbard localization. Nonetheless, the
counterflow low-frequency dynamics of the two components may survive. We
consider corresponding effective Hamiltonians for the three classes of the
two-species insulators-- fermion-fermion, boson-boson, and boson-fermion type--
and reveal the conditions when the resulting groundstate supports
super-counter-fluidity (SCF) of the two components. Alternative groundstates
are found to be phase-segregated states. We emphasize a crucial role of
breaking the isotopic symmetry between the species for realizing the SCF phase.
