  We present an obstruction theoretic inductive construction of intersection
space pairs, which generalizes Banagl's construction of intersection spaces for
arbitrary depth stratifications. We construct intersection space pairs for
pseudomanifolds with compatible trivial structures at the link fibrations; this
includes the case of toric varieties. We define intersection space complexes in
an axiomatic way, similar to Goresky-McPherson axioms for intersection
cohomology. We prove that if the intersection space exists, then the
pseudomanifold has an intersection space complex whose hypercohomology recovers
the cohomology of the intersection space pair. We characterize existence and
uniqueness of intersection space complexes in terms of the derived category of
constructible complexes. We show that intersection space complexes of algebraic
varieties lift to the derived category of Mixed Hodge Modules, endowing
intersection space cohomology with a Mixed Hodge Structure. We find classes of
examples admitting intersection space complex, and counterexamples not
admitting them; they are in particular the first examples known not admitting
Banagl intersection spaces. We prove that the (shifted) Verdier dual of an
intersection space complex is an intersection space complex. We prove a generic
Poincare duality theorem for intersection space complexes.
