  In this series of papers, we propose a theory of enumerative invariants
counting self-dual objects in self-dual categories. Ordinary enumerative
invariants in abelian categories can be seen as invariants for the structure
group $\mathrm{GL} (n)$, and our theory is an extension of this to structure
groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$. Examples of our invariants
include invariants counting principal orthogonal or symplectic bundles, and
invariants counting self-dual quiver representations.
  In the present paper, we take the motivic approach, and define our invariants
as elements in a ring of motives. We also extract numerical invariants by
taking Euler characteristics of these elements. We prove wall-crossing formulae
relating our invariants for different stability conditions. We also provide an
explicit algorithm computing invariants for quiver representations, and we
present some numerical results.
