Two ways to go about this.
Create your box from 3D layers at comp center, add a camera, Add an Orbit Null, rotate the orbit null in Y. This sounds kind of like what you did except that your box is not at comp center or the orbit null is not at the geometric center of your box.
Second, Create your box and put a 3d null at the geometric center of the box. Make the 3D null the parent of all 6 sides of your box. Animate the Y rotation of the null.
The second option has a few advantages. It's easier to create a 3D world to fly your box through and you can easily animate the camera position to push in or pull back from the box.
An easy way to create a box at comp center is to drag all sides of the box to comp center, make all 6 layers 3D layers, press the A key to reveal the anchor point for each layer.
If the vertical sides of the box are all 200 pixels wide and 400 pixels tall then the anchor point for these layers will be 100, 200, 0. To arrange them in a cube set the anchor point for the six sides to 100, 200, 100. With the anchor point set to 100 in Z you rotate side 1 90º side 2 180º and side 3 270º This will create a box exactly at comp center.
If you have top and bottom layers that are 200 X 200 and the sides are 400 pixels tall then set the anchor point for the top and bottom to 100, 100, 100 and rotate the top layer 90º in X and the bottom layer 180º in x.
Add a new null to the comp, make it 3D and then parent all 6 layers that make up your box to the null, lock the side and top layers, and animate the null...
Here's a project file for you to play with. (CS6)
OK, I followed your advice and learned all about null objects and parenting to null objects.
Brilliant! It works just like I had envisioned!
Thank you very much for your response. There is an awful lot to learn in there and I'm getting there a bit at a time.
Mike in Ottawa