  We apply an Osterwalder-Seiler version of twisted mass QCD to a study of the
$B_K$ parameter, in which three of the four quark fields making up the relevant
$\Delta S =2$ operator are maximally twisted with the same twist angle, while
the fourth one has a twist angle of opposite sign. It is known that this setup
ensures automatic improvement of the bare $K^0$-$\overline K^0$ operator matrix
element and multiplicative renormalization of the $\Delta S =2$ operator, at
the price of breaking the $K^0$-$\overline K^0$ mass degeneracy by
discretization effects. As a result, two dominant systematic errors of the $B_
K$ determination with Wilson fermions are kept under control. With the Clover
term included in the fermion action, we perform a feasibility study and find,
in the quenched approximation, a significant improvement of the scaling
behaviour of $B_K$, compared to earlier standard tmQCD determinations.
Moreover, we study in detail the $K^0$-$\overline K^0$ mass splitting that
characterizes this approach and confirm that, in the presence of the Clover
term, it is greatly reduced in a maximally twisted theory.
