  In the present paper we have developed a concept of parallel ordinary (O) and
mirror (M) worlds. We have shown that in the case of a broken mirror parity
(MP), the evolutions of fine structure constants in the O- and M-worlds are not
identical. It is assumed that E_6-unification inspired by superstring theory
restores the broken MP at the scale \sim 10^{18} GeV, what unavoidably leads to
the different E_6-breakdowns at this scale: E_6 \to SO(10)\times U(1)_Z - in
the O-world, and E'_6 \to SU(6)'\times SU(2)'_Z - in the M-world. Considering
only asymptotically free theories, we have presented the running of all the
inverse gauge constants \alpha_i^{-1} in the one-loop approximation. Then a
`quintessence' scenario is discussed for our model of accelerating universe.
Such a scenario is related with an axion (`acceleron') of a new gauge group
SU(2)'_Z which has a coupling constant g_Z extremely growing at the scale
\Lambda_Z\sim 10^{-3} eV.
