Spherical data appear in various applications, including cosmology and the Earth sciences. A standard statistical model for such data employs isotropic spherical random fields. The isotropy assumption may be unrealistic for many real-world datasets. Also, in various applications such as cosmology, only a single realisation of the field is observed, making isotropy impossible to assess through repeated sampling. This paper considers an alternative anisotropic model, referred to as diagonal anisotropy. To test isotropy against diagonal anisotropy, we develop score and cumulative-sum-type tests. Their asymptotic properties and Monte Carlo-based alternatives suitable for moderate sample sizes are established. The performance of the proposed methods is illustrated by numerical studies via applications to simulated data and to actual Planck cosmic microwave background radiation observations. The tests are also applied to three Planck SMICA cosmic microwave background radiation maps and demonstrate how results can vary for different data releases and map construction procedures.

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