A filter chain is a composition.

And composition does not commute. crop=1080:1080:420:0,scale=1080:1920 on a 1920×1080 frame gives a 1080×1920 picture made of a 1080×1080 region at (420,0). The same two filters the other way round does not run at all — after the scale the frame is 1080×1920 and the crop wants 1080×1080 at y 0…1080, which runs off it.

The engine is free and needs no account. Paste the -vf string or the whole ffmpeg command. The smallest rectangle anywhere in the chain is the ceiling on everything after it: a chain through 960×540 that delivers 1920×1080 ships four pixels for every one it had, so 75% of what goes out was invented by a scaler — and a crop is a bottleneck too, so cropping a quarter of the frame and delivering full size invents exactly the same share. And composing the chain backwards says which rectangle of the original frame the output is made of, and where it sat: the question people actually have, which a filter string never answers. Every geometry step is an affine map on coordinates plus a rectangle, so the whole chain is one map and its inverse is exact. The grids are not suggestions either — in 4:2:0 an odd crop offset shifts luma against chroma by half a chroma sample, and no decoder can undo it. Every lane ships a worked example that costs nothing. Writing the report needs credits.