  We show that Tim Cochran's invariants $\beta^i(L)$ of a $2$-component link
$L$ in the $3$--sphere can be computed as intersection invariants of certain
2-complexes in the $4$--ball with boundary $L$. These 2-complexes are special
types of twisted Whitney towers, which we call {\em Cochran towers}, and which
exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation
of the $\beta^i$ invariants for all $i\leq k$, i.e. simultaneous extraction of
invariants from a Whitney tower at multiple orders. This is in contrast with
the order $n$ Milnor invariants (requiring order $n$ Whitney towers) and
consistent with Cochran's result that the $\beta^i(L)$ are integer lifts of
certain Milnor invariants.
