Linux 3.13.0-83-generic #127-Ubuntu SMP Fri Mar 11 00:25:37 UTC 2016 x86_64 x86_64 x86_64 GNU/Linux
Linux 3.13.0-83-generic
Linux-x86_64
nVidia GeForce 8500 GT

Blender 2.77 22a2853

3DConnexion
should be 6 Axis
has 5 Axis

What ever I tried in the last week, I cannot get running a correct z-Axis down and up. Knob down produces an Up-direction and vice versa.
If I use the Menu (Mouse-Knob right) I can invert the axis. Do so, I get the right-left-direction as well as incline corrected, but NOT at the same time.
If I do an invert one of the exis, the other one jumps back.
For example :
Invert the X-Axis, the Y-Axis is false
Invert the Y-Axis, the X-Axis is false
The Z-Axis I cannot invert as described above.

In Freecad there I have the same 3DConnexion in Use.
Here I can ivert the axis, as I will, nothing jumps back to false.
There are also sliders to get the best speed for movements.

I think, the best way would be, to take the code of Freecad inside to blender, because it is a running well system, how anyone can do that for 3DConnexion. It is running very well now for a long time without any issues. Thats my experience.
And - it is very easy to use to make correct settings.

Please take a look for "Tools" --> Userdefined --> Spaceball-Movement, here you can see the Mask, how 3DConnexion can set.
All these in FreeCAD !

Hope, anyone can find the reason for that strange behavior. In this wise nobody can work a long time with a 3D-Mouse. Its like a fly out of control.