  Within its traditional range of perversity parameters, intersection
cohomology is a topological invariant of pseudomanifolds. This is no longer
true once one allows superperversities, in which case intersection cohomology
may depend on the choice of the stratification by which it is defined.
Topological invariance also does not hold if one allows stratifications with
codimension one strata. Nonetheless, both errant situations arise in important
situations, the former in the Cappell-Shaneson superduality theorem and the
latter in any discussion of pseudomanifold bordism. We show that while full
invariance of intersection cohomology under restratification does not hold in
this generality, it does hold up to restratifications that fix the the top
stratum.
