Depends on several things: what exporter you're using (if it's half-well-done it should do axis conversions for you), how your game engine arranges the axes etc.
Don't use orthogonal views to analyse axes. In 3DS Max they do change the axes arrangement (i.e like you realized yourself, in Perspective view Z points up, in side\front views it's Y that points up).
It's so this orthogonal view works like a cartesian system: Y points up, X points sideways and Z points deep.
Anyway, you want to reset pivots or gizmos. You do that in the Hierarchy Panel:
Head for that Hierarchy Panel, click the toggle button "Affect Pivot Only" and now you can transform your object's pivot in any way you want (using the Move, Scale and Rotate tools). In your case, you will rotate\mirror it around so you get the axis alignment you want for your character. I suggest you turn on the Angle Snap toggle button so your rotations are nice rounded values (it's the magnet with the angle symbol button).
When you have all this procedure figured out fine, come back here and we can script a tool that automates this process: so you can select a mesh, use the tool and have its pivot be arranged in the necessary ways automatically (making your own custom tools for things like this gives you plenty more speed when working).