  We study $N=1$ supersymmetric $SO(10)$ gauge theory with a field in the
spinorial representaition and $N_f$ ($\leq8$) fields in the defining
representation. It is shown that this theory for $N_f=7,8$ has a dual
description, which is $N=1$ supersymmetric $SU(N_f-5)$ gauge theory. Its matter
content for $N_f=7$ is different {}from the one for $N_f=8$; for $N_f=7$, it
contains $8$ fields in the anti-fundamental representation. For $N_f=8$, a
rank-$2$ symmetric tensor and one field in the fundamental representation
appears in addition to them. This duality connects along the flat direction to
the duality between chiral and vector gauge theory found by Pouliot.
