Okay, I’ll bite. I have had enough problems in the past with the BGE, linear algebra and matrix transformations, so I want to subscribe to this thread.
I played with it systematically and there is something that is not quite right, though I am not sure what it is. Putting the model into top view and rotating in the global Z-axis
Rotation…screen direction of local Y-axis vector…direction of Y-axis, (or forward,) vector
0 degrees :…(0,1,0)…(0,1,0)
45 degrees :…(-0.7,0.7,0)…(0.7,0.7,0)
90 degrees :…(-1,0,0)…(1,0,0)
135 degrees :…(-0.7,-0.7,0)…(0.7,-0.7,0)
180 degrees :…(0,-1,0)…(0,-1,0)
225 degrees :…(0.7,-0.7,0)…(-0.7,-0.7,0)
270 degrees :…(1,0,0)…(-1,0,0)
315 degrees :…(0.7,0.7,0))…(-0.7,0.7,0)
360 degrees :…(0,1,0)…(0,1,0)
Noting that where the thing appears to be correct, the x coordinate is zero, I would hazard a guess and say that x coordinate of the forward vector is always negative of what it should be?
The real question is why?
Edit:
@agoose77 when you look in the top view, rotating 180 degrees in the local Y-axis, is effectively multiplying the X coordinate of the local Y-axis vector by -1.