The manifold hypothesis suggests that data lies on manifolds with smaller intrinsic dimension (ID) than their ambient dimension. However there is no empirical agreement on the estimates for ID from different estimators for realistic datasets. Thus it is important to test ID estimators (IDEs) with targeted stressors. In this work, we consider the role of anisotropy. To this end, we propose HomID, a collection of homogeneous spaces with anisotropic embedding, for benchmarking ID estimators. We observe that methods that perform well on standard benchmarks systematically degrade on HomID under identical resource allocation. We further observe that anisotropic distortion of such benchmarks also results in performance degradation. Finally, we demonstrate that controlled anisotropic distortions induce systematic shifts in the distributions on which these methods rely, providing a concrete mechanism for the resulting estimation errors in two particular IDEs. The code for the project is available at https://github.com/a-writes-code/homid

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