  This paper describes the relationship between the first non-vanishing Milnor
invariants of a classical link and the intersection invariant of a twisted
Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed
disks bounded by the given link in the 3-sphere together with finitely many
`layers' of Whitney disks.
  The intersection invariant is a higher-order generalization of the
intersection number between two immersed disks in the 4-ball, well known to
give the linking number of the link on the boundary, which measures
intersections among the Whitney disks and the disks bounding the given link,
together with information that measures the twists (framing obstructions) of
the Whitney disks.
  This interpretation of Milnor invariants as higher-order intersection
invariants plays a key role in the classifications of both the framed and
twisted Whitney tower filtrations on link concordance (as sketched in this
paper). Here we show how to realize the higher-order Arf invariants, which also
play a role in the classifications, and derive new geometric characterizations
of links with vanishing Milnor invariants of length less than or equal to 2k.
