Animator Editor adding unwanted curves

I've got a simple little hierarchy of GameObjects for my side-scrolling shmup in Unity 2.6, like so:

Wave 1
|
|- Point 1
|- Point 2
|- Point 3
|- Point 4

Wave 1 has an animation component attached. I want to use the animation system to set up movement patterns for waves of enemies in the game: I'll record movement for Point 1, 2, 3, 4, and then can attach actual enemy prefabs to the points depending on which enemies I want in the wave.

The animation editor is being weird. I can create a new animation clip, but when I expand Point 1's transform and add a keyframe on frame zero, it also adds keyframes to the transforms of points 2 and 3. Particularly weirdly, it doesn't add it for all axes; I get a keyframe on XYZ for Point 1, Y and Z for point 2, and X and Z for point 3.

Why is this happening, and how do I stop it?