  Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis
under the form of covariance matrices or correlation matrices. Several
O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular
the kernel metrics introduced by Hiai and Petz. The class of kernel metrics
interpolates between many classical O(n)-invariant metrics and it satisfies key
results of stability and completeness. However, it does not contain all the
classical O(n)-invariant metrics. Therefore in this work, we investigate
super-classes of kernel metrics and we study which key results remain true. We
also introduce an additional key result called cometric-stability, a crucial
property to implement geodesics with a Hamiltonian formulation. Our method to
build intermediate embedded classes between O(n)-invariant metrics and kernel
metrics is to give a characterization of the whole class of O(n)-invariant
metrics on SPD matrices and to specify requirements on metrics one by one until
we reach kernel metrics. As a secondary contribution, we synthesize the
literature on the main O(n)-invariant metrics, we provide the complete formula
of the sectional curvature of the affine-invariant metric and the formula of
the geodesic parallel transport between commuting matrices for the
Bures-Wasserstein metric.
