Important but rare and subtle processes driving galaxy morphology and star formation may be missed by traditional spiral, elliptical, irregular or Sersic bulge/disc classifications. To overcome this limitation, we use a principal component analysis (PCA) of non-parametric morphological indicators (concentration, asymmetry, Gini coefficient, M_20_, multimode, intensity and deviation) measured at rest-frame B band (corresponding to HST/WFC3 F125W at 1.4<z<2) to trace the natural distribution of massive (>10^10^M_{sun}_) galaxy morphologies. PCA quantifies the correlations between these morphological indicators and determines the relative importance of each. The first three principal components (PCs) capture ~75 per cent of the variance inherent to our sample. We interpret the first PC as bulge strength, the second PC as dominated by concentration and the third PC as dominated by asymmetry. Both PC1 and PC2 correlate with the visual appearance of a central bulge and predict galaxy quiescence. PC1 is a better predictor of quenching than stellar mass, as good as other structural indicators (Sersic-n or compactness). We divide the PCA results into groups using an agglomerative hierarchical clustering method. Unlike Sersic, this classification scheme separates compact galaxies from larger, smooth protoelliptical systems, and star-forming disc-dominated clumpy galaxies from star-forming bulge-dominated asymmetric galaxies. Distinguishing between these galaxy structural types in a quantitative manner is an important step towards understanding the connections between morphology, galaxy assembly and star formation.