  We present several theories of four-dimensional gravity in the Plebanski
formulation, in which the tetrads and the connections are the independent
dynamical variables. We consider the relation between different versions of
gravitational theories: Einstenian, dual, 'mirror' gravities and gravity with
torsion. According to Plebanski's assumption, our world, in which we live, is
described by the self-dual left-handed gravity. We propose that if the Mirror
World exists in Nature, then the 'mirror gravity' is the right-handed
anti-self-dual gravity with broken mirror parity. Considering a special version
of the Riemann--Cartan space-time, which has torsion as additional geometric
property, we have shown that in the Plebanski formulation the ordinary and dual
sectors of gravity, as well as the gravity with torsion, are equivalent. In
this context, we have also developed a 'pure connection gravity' -- a
diffeomorphism-invariant gauge theory of gravity. We have calculated the
partition function and the effective Lagrangian of this four-dimensional
gravity and have investigated the limit of this theory at small distances.
