The study of stable/unstable regular states in high-dimensional physical systems is of paramount importance since they organise the system dynamics by shaping the phase space. Understanding the phase space geometry and the topology of the different attractors is then fundamental for explaining and forecasting global dynamics. With this aim, theoretical and experimental efforts have been made in the analysis of global bifurcations, i. e. sudden changes of the state topology which occur globally in the phase space when some input parameter is varied. One of these types of bifurcations is called “gluing” since, for the most simple case, there are two stable periodic orbits (PO) that glue together at the critical parameter to form a single PO. There are however few examples for which the glued states are invariant tori (quasiperiodic orbits). As previously known on a simple model, the symmetries present in the system have consequences for gluing bifurcations, allowing collisions of multiple regular states rather than a single collision. By analysing the spatial symmetries and the fundamental temporal frequencies of the flow, we describe a new gluing bifurcation scenario for the creation/destruction of three-tori in high-dimensional dissipative dynamical systems, as is the case of the magnetised spherical Couette (MSC) problem. This study constitutes the first example of global bifurcation phenomena occurring in a system capturing the essential features -rotation, spherical geometry, and magnetic fields- of geo-and astrophysical flows.