Canonical models of single cell migration describe alignment between the Golgi (and microtubule organising centre) and nucleus relative to the direction of cell motility. Yet little quantitative data exists to support this fundamental model during random single cell migration. To define the relationship between cell polarity and motion, we imaged and quantitatively analysed spatiotemporal variations in cell motion, polarity and morphology in individual randomly migrating cells. This was based on ~250,000 live cell observations recorded over a two-dimensional condition array wherein intra-cellular tension (modulated via ROCK-signalling) and extra-cellular matrix (ECM) ligand density were progressively co-varied. This revealed a complex, highly plastic relationship that is subject to non-linear regulation. Under most (though not all) conditions, cells tend toward adoption of forward polarity (Golgi anterior to nucleus), though backward polarity is common. Cells with forward polarity are relatively ordered and responsive to regulation, while backward polarity reflects disorder. The balance between these orientations is regulated by both tension and adhesion levels, though the impacts of these regulations are inter-dependent and plastic, e.g. the effects of tension can be inverted by changing adhesion levels, and vice versa. Internal dependencies are also non-linear, for example, forward polarity alignment correlates with migration speed, but only to a maximum, after which increases in speed destabilise alignment. Finally, we reveal for the first time that the angles of motion and polarity both oscillate, in a coupled manner at a frequency of ~100 minutes. This suggests the existence of a previously undetected bi-directional motion-polarity feedback mechanism.