The first part of this paper completes the classification of Whitney towers
in the 4-ball that was started in three related papers. We provide an algebraic
framework allowing the computations of the graded groups associated to
geometric filtrations of classical link concordance by order $n$ (twisted)
Whitney towers in the 4-ball. Higher-order Sato-Levine invariants and
higher-order Arf invariants are defined and shown to be the obstructions to
framing a twisted Whitney tower. In the second part of this paper, a general
theory of quadratic forms is developed and then specialized from the
non-commutative to the commutative to finally, the symmetric settings. The
intersection invariant for twisted Whitney towers is shown to be the universal
symmetric refinement of the framed intersection invariant.