  Given a closed symplectic manifold, we construct invariants which count (a)
closed rational pseudoholomorphic curves with prescribed cusp singularities and
(b) punctured rational pseudoholomorphic curves with ellipsoidal negative ends.
We prove an explicit equivalence between these two frameworks, which in
particular gives a new geometric interpretation of various counts in symplectic
field theory. We show that these invariants encode important information about
singular symplectic curves and stable symplectic embedding obstructions. We
also prove a correspondence theorem between rigid unicuspidal curves and
perfect exceptional classes, which we illustrate by classifying rigid
unicuspidal (symplectic or algebraic) curves in the first Hirzebruch surface.
