This paper delves into the origins and specificity of the unique stable spin textures (360° closed loop domain walls and skyrmions) observed in exchange-biased systems, with either in-plane or out-ofplane magnetic anisotropy. In the case of skyrmions, which are nanometer-sized bubbles consisting of closed-loop 180° walls in perpendicularly-magnetized media, the stability of these spin textures arises from the existence of Dzyaloshinskii-Moriya Interactions (DMI). These interactions induce chirality of the domain walls, yielding to some extent a so-called topological protection. More complex structures such as skyrmoniums have been observed, consisting of closed loop 360° walls. Recently, skyrmions formed in the absence of an applied external magnetic field have been stabilized in exchange biased out-of-plane magnetized systems. About two decades ago, another type of stable spin-textures were observed in exchange biased systems, with in-plane magnetization, in particular in the pinned reference layer of spin-valves. These textures consist of 360°-domain-wall rings, the stability of which arises from the easy-plane shape anisotropy of these layers. In this paper, we compare these spintextures and highlight the similarities and differences in their formation, structure and origin of their stability.