  Motivated by suggestions that binaries with almost equal-mass components
("twins") play an important role in the formation of double neutron stars and
may be rather abundant among binaries, we study the stability of synchronized
close and contact binaries with identical components in circular orbits. In
particular, we investigate the dependency of the innermost stable circular
orbit on the core mass, and we study the coalescence of the binary that occurs
at smaller separations. For twin binaries composed of convective main-sequence
stars, subgiants, or giants with low mass cores (M_c <~0.15M, where M is the
mass of a component), a secular instability is reached during the contact
phase, accompanied by a dynamical mass transfer instability at the same or at a
slightly smaller orbital separation. Binaries that come inside this instability
limit transfer mass gradually from one component to the other and then coalesce
quickly as mass is lost through the outer Lagrangian points. For twin giant
binaries with moderate to massive cores (M_c >~0.15M), we find that stable
contact configurations exist at all separations down to the Roche limit, when
mass shedding through the outer Lagrangian points triggers a coalescence of the
envelopes and leaves the cores orbiting in a central tight binary. In addition
to the formation of binary neutron stars, we also discuss the implications of
our results for the production of planetary nebulae with double degenerate
central binaries.
