Roughly speaking, to any space $ M$ with perfect obstruction theory we associate a space $ N$ with symmetric perfect obstruction theory. It is a cone over $ M$ given by the dual of the obstruction sheaf of $ M$ and contains $ M$ as its zero section. It is locally the critical locus of a function.

More precisely, in the language of derived algebraic geometry, to any quasi-smooth space $ M$ we associate its $ (\!-\!1)$-shifted cotangent bundle $ N$.

By localising from $ N$ to its $ \mathbb{C}^*$-fixed locus $ M$ this gives five notions of a virtual signed Euler characteristic of $ M$:

The Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of $ M$ defined using its own obstruction theory,
Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of $ N$ to $ M$,
Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of $ N$ to $ M$,
Kiem-Li's cosection localisation of the virtual cycle of $ N$ to $ M$,
$ (-1)^{\textrm {vd}}$ times by the topological Euler characteristic of $ M$.
Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.