It is well-known that theta characteristics on smooth plane curves over a field
of characteristic different from two are
in bijection with certain smooth complete intersections of three quadrics.
In this talk, we generalize this bijection to possibly singular hypersurfaces of any dimension
over arbitrary fields including those of characteristic two.
It is accomplished in terms of linear orbits of tuples of symmetric matrices
instead of smooth complete intersections of quadrics.

As an application of our methods, we give a description of the
projective automorphism groups of complete intersections of quadrics
generalizing Beauville's results.