  We consider a class of symmetry hypothesis testing problems including testing
isotropy on $\mathbb{R}^d$ and testing rotational symmetry on the hypersphere
$\mathcal{S}^{d-1}$. For this class, we study the null and non-null behaviors
of Sobolev tests, with emphasis on their consistency rates. Our main results
show that: (i) Sobolev tests exhibit a detection threshold (see Bhattacharya,
2019, 2020) that does not only depend on the coefficients defining these tests;
and (ii) tests with non-zero coefficients at odd (respectively, even) ranks
only are blind to alternatives with angular functions whose $k$th-order
derivatives at zero vanish for any $k$ odd (even). Our non-standard asymptotic
results are illustrated with Monte Carlo exercises. A case study in astronomy
applies the testing toolbox to evaluate the symmetry of orbits of long- and
short-period comets.
