The hyperelliptic Torelli group, SI(S_g), is the subgroup of the mapping class group consisting of those elements that commute with a fixed hyperelliptic involution ι and act trivially on the homology of the surface S_g. The group SI(S_g) appears in a variety of contexts, e.g., as a kernel of a Burau representation and as the fundamental group of the branch locus of the period mapping on Torelli space. The main result of this paper is that, for g ≧ 3, we have Aut(SI(S_g)) ≅ SMod^±(S_g)/<ι>, where SMod^±(S_g) is the extended hyperelliptic mapping class group. Our main tool is the symmetric separating curve complex, C_ssep(S_g), and we show that if g ≧ 3, Aut(C_ssep(S_g)) ≅ SMod^±(S_g)/<ι>. Another key ingredient is an algebraic characterization of Dehn twists about symmetric separating curves. These results are analogous to results of Ivanov, Farb-Ivanov, and Brendle-Margalit for the mapping class group, the Torelli group, and the Johnson kernel.