Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T₅ spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theorem (Theorem 1) that applies to all such ω₁-compact spaces and another (Theorem 4) to all such spaces of Lindelöf number ≤ ℵ₁. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact subspaces in certain continuous preimages of ω₁. It also exposes (Theorem 2) the fine structure of perfect preimages of ω₁ which are T₅ and hereditarily collectionwise Hausdorff. In these theorems, "T₅ and hereditarily collectionwise Hausdorff" is weakened to "hereditarily strongly collectionwise Hausdorff." Corollaries include the consistency, modulo large cardinals, of every hereditarily strongly collectionwise Hausdorff manifold of dimension > 1 being metrizable. The concept of an alignment plays an important role in formulating several of the structure theorems.