Geometrical heterogeneities along layer interfaces play a key role in determining the geometries of folds developed during shortening of competent layers. We present a series of numerical simulations to investigate the influence of initial sinusoidal perturbations on the folding of single layers. Models consist of a competent viscous single layer embedded in a softer matrix, with the layer orientated parallel to the shortening direction. We first generalize the wide spectrum of sinusoidal perturbations accounting for asymmetries along and across a competent single layer, using two parameters: transversal asymmetry (A′) and longitudinal asymmetry (φ). These two parameters allow the transition between classical fold shapes and pinch and swell geometries to be studied. The parameter A′ describes the development of fold hinges with different geometries between the upper and lower layer interfaces, and abnormal curvatures between the outer and inner arcs of fold hinges. The parameter φ induces a strong polarity on the folds, with a systematic preferred orientation of the pinch and swell regions of the layer, even if there is no shear component parallel to the layer. Our results demonstrate the importance of structural inheritance on the resulting fold geometries, and suggest that caution must be taken when using certain types of asymmetrical folds as strain markers and kinematic indicators.