  We study a class of combinatorially-defined polynomial ideals which are
generated by minors of a generic symmetric matrix. Included within this class
are the symmetric determinantal ideals, the symmetric ladder determinantal
ideals, and the symmetric Schubert determinantal ideals of A. Fink, J.
Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a
Kazhdan-Lusztig ideal of A. Woo and A. Yong; that is, it is the
scheme-theoretic defining ideal of the intersection of a type C Schubert
variety with a type C opposite Schubert cell, appropriately coordinatized. The
Kazhdan-Lusztig ideals that arise are exactly those where the opposite cell is
$123$-avoiding. Our main results include Gr\"obner bases for these ideals,
prime decompositions of their initial ideals (which are Stanley-Reisner ideals
of subword complexes) and combinatorial formulas for their multigraded Hilbert
series in terms of pipe dreams.
