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 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