What is the medieval term for North African Muslim rule in Spain (early period) linked to Damascus?

The story behind the answer

The early Muslim rulers of al-Andalus traced their legitimacy to the Umayyad dynasty, which led the caliphate from Damascus (661–750). After the Abbasids overthrew them, the Umayyad prince Abd al‑Rahman I escaped across North Africa and established the Emirate of Córdoba in 756, later proclaimed a caliphate in 929—preserving a Damascene legacy in Iberia.

By contrast, the Aghlabids were a vassal dynasty in Ifriqiya (modern Tunisia) from 800–909, the Fatimids were an Isma’ili caliphate centered in Mahdia and later Cairo (909–1171), and the Abbasids ruled from Baghdad after 750. None of these governed early al‑Andalus, which was defined by its Umayyad foundation.