  We propose a scheme to realize a pseudospin-$1/2$ model of the
$^{1}\Sigma(v=0)$ bialkali polar molecules with the spin states corresponding
to two sublevels of the first excited rotational level. We show that the
effective dipole-dipole interaction between two spin-$1/2$ molecules couples
the rotational and orbital angular momenta and is highly tunable via a
microwave field. We also investigate the ground state properties of a
spin-$1/2$ molecular condensate. A variety of nontrivial quantum phases,
including the doubly-quantized vortex states, are discovered. Our scheme can
also be used to create spin-$1$ model of polar molecules. Thus, we show that
the ultracold gases of bialkali polar molecules provide a unique platform for
studying the spinor condensates of rotating molecules.
