In this survey we review classical and recently proposed Riemannian metrics and interpolation schemes on the space of symmetric positive definite (SPD) matrices. We perform simulations that illustrate the problem of tensor fattening not only in the usually avoided Frobenius metric, but also in other classical metrics on SPD matrices such as the Wasserstein metric, the affine invariant/Fisher Rao metric, and the log Euclidean metric. For comparison, we perform the same simulations on several recently proposed frameworks for SPD matrices that decompose tensors into shape and orientation. In light of the simulation results, we discuss the mathematical and qualitative properties of these new metrics in comparison with the classical ones. Finally, we explore the nonlinear variation of properties such as shape and scale throughout principal geodesics in different metrics, which affects the visualization of scale and shape variation in tensorial data. With the paper, we will release a software package with Matlab scripts for computing the interpolations and statistics used for the experiments in the paper (Code is available at https://sites.google.com/site/aasaferagen/home/software).