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Try experimenting and seeing how it looks, by draining fuel tanks in the VAB, while watching the "CoM" vs "CoT"
One of the things that make spaceplanes so difficult is that in order to be properly balanced for low speed flight your center of mass should be just a bit forward of your center of lift.
Your center of lift usually provides a lot of drag at high speed and your center of thrust is behind it pushing hard against all that drag. Since your center of mass is so close to your center of drag, you don't get much directional stability.
Add to that how most planes are not vertically symmetrical and you've got a problem.
If you put air intakes or a vertical stabalizer on the top you're adding drag above the center of thrust. If you put your wings below the centerline of the craft, you're adding drag below the COT.
And if all that's not bad enough, drag works differently once you're supersonic.
Spaceplanes are hard. If you've got versions that work to LKO you're doing pretty good.
which i guess is not enough to keep that extremely back-heavy design stable.
you can try and add some more reaction wheels and see if they are enough to keep the plane controllable. shifting the mass of the engines closer to the mid section of the plane would probably be a better design, though. the engines attached at the end of the main stack/fuselage can't be moved much, but the radially attached whiplash and nerv stacks (along with most of the wings) could be moved from the back to the center of the plane to make it a lot more inherently stable.
balancing an SSTO for both full and empty tanks is a delicate task. there are many techniques that can help, but simply moving some of those heavy engines from the back to the mid should help a lot already.
http://steamcommunity.com/sharedfiles/filedetails/?id=723831843
http://steamcommunity.com/sharedfiles/filedetails/?id=723831859
I have to agree with what mkunz2 is saying.
Planes flip in atmosphere if their center of mass moves behind their center of lift. You can see where these are by clicking on those buttons in the lower left side of the screen, to the right of the cost and above the symmetry controls. You can also adjust how full the tanks are in the VAB and SPH by cllicking on the tanks.