Representational similarity analysis and related methods have become standard tools for comparing the internal geometries of neural networks and biological systems. These methods measure what is represented, the alignment between two representational spaces, but not whether that structure is robust. We introduce geometric stability, a distinct dimension of representational quality that quantifies how reliably a representation's pairwise distance structure holds under perturbation. Our metric, Shesha, measures self-consistency through split-half correlation of representational dissimilarity matrices constructed from complementary feature subsets. A key formal property distinguishes stability from similarity: Shesha is not invariant to orthogonal transformations of the feature space, unlike CKA and Procrustes, enabling it to detect compression-induced damage to manifold structure that similarity metrics cannot see. Spectral analysis reveals the mechanism: similarity metrics collapse after removing the top principal component, while stability retains sensitivity across the eigenspectrum. Across 2463 encoder configurations in seven domains -- language, vision, audio, video, protein sequences, molecular profiles, and neural population recordings -- stability and similarity are empirically uncorrelated ($ρ=-0.01$). A regime analysis shows this independence arises from opposing effects: geometry-preserving transformations make the metrics redundant, while compression makes them anti-correlated, canceling in aggregate. Applied to 94 pretrained models across 6 datasets, stability exposes a "geometric tax": DINOv2, the top-performing model for transfer learning, ranks last in geometric stability on 5/6 datasets. Contrastive alignment and hierarchical architecture predict stability, providing actionable guidance for model selection in deployment contexts where representational reliability matters.