  Exotic aromatic B-series were originally introduced for the calculation of
order conditions for the high order numerical integration of ergodic stochastic
differential equations in $\mathbb{R}^d$ and on manifolds. We prove in this
paper that exotic aromatic B-series satisfy a universal geometric property,
namely that they are characterised by locality and orthogonal-equivariance.
This characterisation confirms that exotic aromatic B-series are a fundamental
geometric object that naturally generalises aromatic B-series and B-series, as
they share similar equivariance properties. In addition, we classify with
stronger equivariance properties the main subsets of the exotic aromatic
B-series, in particular the exotic B-series. Along the analysis, we present a
generalised definition of exotic aromatic trees, dual vector fields, and we
explore the impact of degeneracies on the classification.
