  We have developed a concept of parallel existence of the ordinary (O) and
mirror (M), or shadow (Sh) worlds. E_6 unification, inspired by superstring
theory, restores the broken mirror parity at the scale ~ 10^18 GeV. With the
aim to explain the tiny cosmological constant, we consider the breakings: E_6
-> SO(10) X U(1)_Z - in the O-world, and E'_6 -> SU(6)' X SU(2)'_\theta - in
the Sh-world. We assume the existence of shadow \theta-particles and the low
energy symmetry group SU(3)'_C X SU(2)'_L X SU(2)'_\theta X U(1)'_Y in the
shadow world, instead of the Standard Model. The additional non-Abelian
SU(2)'_\theta group with massless gauge fields, "thetons", has a macroscopic
confinement radius 1/\Lambda'_\theta. The assumption that
\Lambda'_\theta\approx 2.3 X 10^-3 eV explains the tiny cosmological constant
given by recent astrophysical measurements. Searching for the Dark Matter (DM),
it is possible to observe and study various signals of theta-particles.
