Hard Light and Overlay are commuted versions of the same formula: Hard Light treats the blend layer as Overlay treats the base layer

Hard Light: f(a,b) = 2ab if b<0.5, else 1-2(1-a)(1-b). This is identical to Overlay but with the condition checking b (the top layer) instead of a (the base). This means Hard Light treats the blend layer the same way Overlay treats the base layer. The article states: ‘Hard Light affects the blend layer’s relationship to the base layer in the same way Overlay affects the base layer’s relationship to the blend layer.’ They are called ‘commuted blend modes.’ Practically: applying Overlay and then swapping layers equals applying Hard Light.

Examples

Overlay of layer A over B = Hard Light of B over A. Both produce the same S-curve contrast effect but the which-layer-controls-the-split differs.

Assessment

Without computing, predict: if you switch the base and blend layers in an Overlay composite, what mode does the result equal? Verify with the formulas.