  We examine the rotational states of a pair of polar $^2\Sigma$ molecules
subject to a uniform magnetic field. The electric dipole-dipole interaction
between the molecules creates entangled pair-eigenstates of two types. In one
type, the Zeeman interaction between the inherently paramagnetic molecules and
the magnetic field destroys the entanglement of the pair-eigenstates, whereas
in the other type it does not. The pair-eigenstates exhibit numerous
intersections, which become avoided for pair-eigenstates comprised of
individual states that meet the selection rules $\Delta J_{i}=0,\pm 1$, $\Delta
N_{i}=0,\pm 2$, and $\Delta M_{i}=0,\pm 1$ imposed by the electric
dipole-dipole operator. Here $J_{i}$, $N_{i}$ and $M_{i}$ are the total,
rotational and projection angular momentum quantum numbers of molecules $i=1,2$
in the absence of the electric dipole-dipole interaction. We evaluate the
mutual alignment of the pair-eigenstates and find it to be independent of the
magnetic field, except for states that undergo avoided crossings, in which case
the alignment of the interacting states is interchanged at the magnetic field
corresponding to the crossing point. We present an analytic model which
provides ready estimates of the pairwise alignment cosine that characterises
the mutual alignment of the coupled rotors.
