Incorrect Lighting on Simple Entity

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I have a small problem with a transparent textured entity:
Example 1: Trees I made with leaf surfaces where TextureFilters "leaves",4 does what it should, and all black pixels are transparent. Hurrah!
Example 2: Fences I made with fence textures where TextureFilters "fence",4 does what it should, and all black pixels are transparent. Hurrah!
BUT -
the lighting (ambient or direct) looks all wrong on the fences!
The fences stay very bright (unlike the trees) even when the ambient and direct lights are taken down.
The only clue that I can think of is that the fence is the only entity I have which is just a plane (two triangles) with the texture mapped on.
The effect is that it looks as if the upper-left and lower-right vertices react correcty to light, but the lower-left and upper-right vertices don't react at all!
Anyone got any ideas, or seen this effect before?