Rational design of the programmable soft matter requires understanding of the effect of a complex metric on shape transformations of thin non-Euclidean sheets. In the present work, we explored experimentally and using simulations how simultaneous or consecutive application of two orthogonal perturbations to thin patterned stimuli-responsive hydrogel sheets affects their three-dimensional shape transformations. The final shape of the sheet is governed by the metric, but not the order, in which the perturbations are applied to the system, and is determined by the competition of small-scale bidirectional stresses. In addition, a new, unexpected transition from a planar state to an equilibrium helical shape of the hydrogel sheet is observed via a mechanism that is yet to be explained.