This article explores hypothetical sundial behaviour on Uranus, where the planet's axis tilt is almost 90 degrees. At solstices, the sun would appear stationary overhead at the pole, and a rod gnomon would cast no shadow. As the sun moves, shadows would form circles, indicating a sidereal day, with significant changes in day length at solstices, leading to a "missing" solar day.
Dials: Unusual,
How Sundials Work,
Mathematics of Dialling