Asymmetrical disruptions may occur during ITER operation and they may be accompanied by large sideways forces and rotation of the asymmetry. This is of particular concern because resonance of the rotating asymmetry with the natural frequencies of the vacuum vessel (and other in-vessel components) could lead to large dynamic amplification of the forces. A significant fraction of non-mitigated JET disruptions have toroidally asymmetric currents that flow partially inside the plasma and partially inside the surrounding vacuum vessel ('wall'). The toroidal asymmetries (otherwise known as the appearance of 3D structures) are clearly visible in the plasma current (Ip) and the first plasma current moments. For the first time we present here the asymmetries in toroidal flux measured by the diamagnetic loops and also propose a physical interpretation. The presented data covers the period of JET operation with a C-wall (JET-C from 2005 until late 2009) and with an ITER-like wall (JET-ILW from 2011 until late 2014), during which pick-up coil and saddle loop data at four toroidally orthogonal locations were routinely recorded. The observed rotations of the Ip asymmetries are in the range from −5 turns to +10 turns (a negative value is counted to the negative plasma current). Initial observations on COMPASS of asymmetric disruptions are presented, which are in line with JET data. The whole of the JET-ILW disruption database and the limited number of COMPASS disruptions examined confirm that the development of the toroidal asymmetry precedes the drop to unity of q95. It is shown that massive gas injection (MGI), which is routinely used to mitigate disruptions, significantly reduces the Ip asymmetries in JET. However, MGI produces fast plasma current quench and consequently high vessel eddy currents, which expose the machine to additional stresses. The effect of the large gas quantity used during the injection is of particular concern as well.