  A 2-component oriented link in $S^3$ is called weakly doubly slice if it is a
cross-section of an unknotted sphere in $S^4$, and strongly doubly slice if it
is a cross-section of a 2-component trivial spherical link in $S^4$. We give
the first example of 2-component boundary links which are weakly doubly slice
but not strongly doubly slice. We also introduce a new invariant $g_{st}$ of
homotopically trivial links that measures the failure of a link from being
strongly doubly slice and that bounds the doubly slice genus $g_{ds}$ from
below. Our examples have arbitrarily large doubly slice genus but satisfy
$g_{st}=1$. We also prove that the Conway-Orson signature lower bound on
$g_{ds}$ is actually a lower bound on $g_{st}$.
