  We find nearly all the exact ground states of a mixture of two species of
spin-1 atoms with both interspecies and intraspecies spin exchanges in absence
of a magnetic field. The quantum phase diagram in the three-dimensional
parameter space and its two-dimensional cross sections are described. The
boundaries where the ground states are either continuous or discontinuous are
determined, with the latter identified as where quantum phase transitions take
place. The two species are always disentangled if the interspecies spin
coupling is ferromagnetic or zero. Quantum phase transitions occur when the
interspecies spin coupling varies between antiferromagtic and zero or
ferromagnetic while the two intraspecies spin couplings both remain
ferromagnetic. On the other hand, by tuning the interspecies spin coupling from
zero to antiferromagnetic and then back to zero, one can circumvent the quantum
phase transition due to sign change of the intraspecies spin coupling of a
single species, which is spin-decoupled with the other species with
ferromagnetic intraspecies spin coupling. Overall speaking, interplay among
interspecies and two intraspecies spin exchanges significantly enriches quantum
phases of spinor atomic gases.
