The SpaceX Grasshopper vehicle (formally a VTVL vehicle) tests have been increasing the height that their tests can climb to before doing the vertical landing. Throughout all these tests it keeps upright.
But in practice, operating as a rocket stage, this vehicle will fall from a substantial height. Before it gets close to the ground, would it keep vertical as it falls, or would it tumble, or fall oriented horizontally?
As the concept video originally released shows, the Grasshopper System will be able to stabilize itself, and land vertically. No doubt there will be some loss of complete control in places, but in the end, the spacecraft must be controlled, and will land vertically. See the YouTube video below:
This makes sense, for a lot of reasons. The primary reason is that the rocket is more stable in a vertical configuration than a horizontal, allowing for more accurate control. No doubt some re-orientation will occur, but as a whole, the system will work better if operating in a vertical configuration.
This is not meant to be cheeky, but if I understand your question correctly, then the answer seems to be contained in the name of the vehicle.
VTVL: Vertical Takeoff Vertical Landing.
I think that even if the Grasshopper had no active flight controls, a large gyroscope would suffice to keep it vertical throughout flight, but the article makes it clear that it does have active flight controls. The Wikipedia article mentioned that:
So in short, I think the answer to your question is: "vertical."
The full flight profile of a VTVL Grasshopper-like vehicle hasn't been shared by SpaceX, but it certainly won't "stay vertical" for the entire flight — assuming that vertical is defined as 90° flight path angle.
Ground launch rockets typically start by flying more or less straight up. Because they are at low angle of attack this keeps lateral aerodynamic forces manageable and it gets the rocket out of the thick lower atmosphere as quickly as possible. But as soon as possible (exactly where depends on all sorts of parameters but above 100,000 ft would be an order of magnitude altitude) they begin what's called a gravity turn. The trajectory begins to turn over and the flight path angle decreases.
So it is likely that the vehicle will have started to turn over by staging and thus will no longer be vertical.
Assuming the vehicle falls ballistically before restarting the main engines for landing, there are really only two high-level options:
Which is best to choose would be a trade buried in the details of the rocket and trajectory design, and which we'll only know the answer to when SpaceX decides to share it.
Either way, the vehicle will have to be properly re-oriented for engine ignition before it got too deep into the atmosphere. Depending on the aerodynamic stability of the vehicle this could be done in part passively or with (or augment with) active control. Because it's only a first stage it will only be going several Mach, and thus probably won't need a thermal protection system as heavy, complex, or expensive as what was used for Shuttle or even the Dragon or Apollo re-entry vehicles.
The booster is probably stable without controlled thrust once the landing legs are deployed, the spaceX description says the engine(s) will be fired 3 times after separation.
The booster will only turn around backwards to direction of flight during the decel and landing phases. It will use aerodynamic drag to reduce the majority of its speed and not use extra fuel.
