  We perform a throughout study of 3+1 dim. SO(3) LGT for any fixed-twist
background. We concentrate in particular on the physically significant trivial
and 1-twist sectors. Introducing a Z(2) monopole chemical potential the 1st
order bulk transition is moved down in the strong coupling region and weakened
to 2nd order in the 4-dim Ising model universality class. In this extended
phase diagram we gain access to a confined phase in every fixed twist sector of
the theory. The Pisa disorder operator is employed together with the Polyakov
loop to study the confinement-deconfinement transition in each sector. Due to
the specific properties of both operators, most results can be used to gain
insight in the ergodic theory, where all twist sectors should be summed upon.
An explicit mapping of each fixed twist theory to effective positive plaquette
models with fixed twisted boundary conditions is applied to better establish
their properties in the different phases.
