A level vial can be made in two ways. The tube can be slightly bent to create an arc or it can be ground barrel shaped. In the case of the former, the vial will be sensitive to axial rotation. This can be observed on older mason's levels. It you tilt the level slightly , it will change the position of the bubble.
For this reason, precision levels have a cross vial and for accurate readings, the surface in the cross direction should be reasonably level. If your vial is one of the former type and it is not installed so the plane of the curve is perfectly perpendicular to the bottom surface of the level, and the surface you are leveling is not perfectly level in the cross direction, you would experience the phenomenon that you see.
Since you have other machinist's levels, I would suggest that you shim up a surface plate on three bosses so it is level in all directions. You should then see no variation in reading when you reverse the level. If you tilt the surface slightly in the cross direction to the measurement line and you then see a variation the vial is not installed correctly.
Note: On their website, Starrett states that their vials a precision ground which would imply that they are the barrel shaped design and therefore should be unaffected by vial rotation. However, there is no guarantee if the vial is of aftermarket manufacture..