I asked this on the Unity Forum, but got only chirping crickets.
But I believe shadow support depends on the shader, and this is using the Spine/Skeleton shader, so maybe I'll have better luck here.
A picture can probably explain it better than words:
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I have a single light in the scene, a point light with a range of 100. That's a unit cube, which is casting a shadow just fine, and on top of it (with his foot slightly embedded in the cube) is my Spine character.
Using a directional light, this character casts shadows just fine. But with the point light, instead of casting shadows, he casts anti-shadows... that is, its only effect is to disrupt shadows cast by other objects. You can see this in the image above; the cube shadow has a foot-shaped hole in it, and the platform's shadow has a cutout in the shape of the character as well.
So... WTF? And is there anything I can do about it? (A directional light just doesn't work well for this scene.)