I was trying to fade shadows out and found animated shadow fade on Blender StackExchange.
In the setup described above, shadow rays are multiplied by a factor, and that's applied to the factor on a Mix Shader that uses 0 for a Diffuse shader and 1 for a Transparent shader. You can animate the amount of shadow casting by animating the multiplication: 0 means cast shadows, and it fades up to 1, where it doesn't cast shadows. I duplicated that in a material that's normally blue, as in the screenshot below.
I found that counterintuitive, and wanted to switch the multiplication value so that 0 meant no shadows, and 1 meant regular full shadows. "No problem! Just swap the Mix Shader inputs so the Transparent is on top, and Diffuse is on bottom! Like the red material in the screenshot below.
That does eliminate the shadow at 0 and show it at 1, but the object is completely transparent. The screenshot below shows a scene with two cubes that would normally cast shadows to their LEFT. The left, blue cube is not casting shadows, with multiply of 1, as expected. The right, red cube casts faded shadows on an inverted factor, as desired, but is unexpectedly completely transparent.
Obviously my understanding is flawed. Can someone explain this effect to me?