Plan irregular structures, whose complex response represents a generalisation of the simpler de-coupled motion ascribed to symmetric buildings, make up a large proportion of the failures during major earthquakes. This paper examines the seismic response scaling of degrading and no-degrading eccentric structures subjected to bidirectional earthquake action and its relationship with the duration of the ground motion by means of dimensional and orientational analyses. Structures with reflectionally symmetric stiffness distribution and mass eccentricity subjected to orthogonal pairs of ideal pulses are considered as the fundamental case. The application of Vaschy–Buckingham's (Formula presented.) -theorem reduces the number of variables governing the peak orthogonal displacements leading to the emergence of remarkable order in the structural response. If orientationally consistent dimensionless parameters are selected, the response becomes self-similar. By contrast, when degradation is introduced, peak inelastic displacements are dramatically affected and the self-similarity in the response is lost immediately after the onset of inelastic deformations. Conversely, if the uniform duration, instead of the period, of the strong motion is adopted as a timescale, a practically self-similar response is observed. This offers unequivocal proof of the fundamental role played by the ground-motion duration in defining the peak displacement response of degrading structures even at small inelastic demands, although its importance increases with increasing deformation levels. Finally, the existence of complete similarities, or similarities of the first kind, are explored and the practical implications of these findings are briefly outlined in the context of real pulse-like ground motions with varying degrees of coherency.