If I understood you correctly, the thing you want to do is "normal" texture mapping. How did you "project" the texture onto your plane? In Mayas terminology (the only one I know) this is referred to as Mapping. In the "Create UVs"'s menu you'll find different mapping methods. For a plane I suggest the "Best Plane Texturing Tool". This works easy: You select the tool, then the faces of your plane and 3 points to define the plane. Now Maya creates UV coordinates for the plane and they are connected with you plane. No matter how you turn it, it will remain the same texture. You'll check the result in the "UV Texture Editor".
But if you are talking about Image Planes (those helpful "virtual" planes, that you use as blueprints for your modelling), things are different. They don't stick to the object by default but to the camera. You can change that behaviour by selecting the camera (In the View panel select your camera by choosing "Select Camera"). You'll find on option "Image Plane" in the section "Image Plane Attributes" in the imagePlane-Tab. There you have "Attached To Camera" and "Fixed".
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