Symmetric Positive Definite (SPD) matrices have been widelyused in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very fewmethodological principles guiding the choice of one particular metric fora given application. Invariance under the action of the affinetransfor-mations was suggested as a principle. Another concept is based on sym-metries. However, the affine-invariant metric and the recently proposedpolar-affine metric are both invariant and symmetric. Comparing thesetwo cousin metrics leads us to introduce much wider families: power-affineand deformed-affine metrics. Within this continuum, we investigate otherprinciples to restrict the family size.