In fact the answer is "yes", the non-chiral type $II$ sugra thoery is called type $II$ A. You can obtain it by dimensional reduction from the $M$-theory sugra in $d=11$. The (massless) spectrum of type $II$ B contains spinor representations of just one chirality (which one is matter of convention), while type $II$ A contains representations of both chiralities.

The fact you say about the $\mathcal{N}=2$ supergauge theory is true in $d=4$, but not in general.

This post imported from StackExchange Physics at 2015-03-12 12:17 (UTC), posted by SE-user Oscar