  We construct new subtle Stiefel--Whitney classes of quadratic forms. These
classes are much more informative than the ones introduced by Milnor. In
particular, they see all the powers of the fundamental ideal of the Witt ring,
contain the Arason invariant and it's higher analogues. Moreover, the new
classes allow to treat the J-invariant of quadrics. This invariant, introduced
in 2005, has been so far completely isolated from characteristic classes. In
addition, our classes allow to describe explicitly the structure of some
motives associated with quadratic forms.
