The turntable orbiting method, while being the preffered one by most users has one big problem in Blender. The fixed rotation axis is always fixed to be the global Z axis. While this works perfectly fine in most cases, it starts to cause problems when the viewport is rolled (e.g. using shift 4 or shift 6). Unlike one might expect, the fixed axis rolls with the viewport but the left/right orbiting directions (whatever you call them) stays fixed. Apart from causing unpredictable behaviour during viewport orbiting, effectively making it impossible to orbit, it also creates some really bad problems. One of these is something akin to gimbal lock, which can be achieved by rolling the viewport by 90 degrees. Trying to orbit the viewport using the mouse will now result in Blender completely acting up, but if you press numpad 2,4,6,8 it confirms that we indeed lose one degree of freedom.
When using the turntable there is a fixed axis around which the viewport always orbits (the Z axis when the viewport hasn’t been rolled).
What happens is that Blender rolls this fixed axis with the viewport, so that when you orbit up/ down it does that using the new rolled fixed axis.
However it does not roll the left/right orbiting directions (whatever you call them) with the viewport, meaning that, in my example, it creates an issue akin to gimbal lock where orbiting up/down will cause the same rotation of the viewport effectively making it lose one degree of freedom.
I don’t know the reason why Blender seems to behave in an unpredictable manner when orbiting with the mouse, but when orbiting with the numpad keys it confirms that we do indeed lose one degree of freedom as both Numpad 4 and 8 both cause an upward rotation of the viewport and both Numpad 2 and 6 cause a downward rotation of the viewport.
So to fix the issue all that needs to be done is to roll the left/right orbiting directions with the viewport and fixed axis.
I consider this behaviour to probably be an oversight by the developers or a result of bad viewport navigation design. Either way it should be fixed as soon as possible.
For more information please see T98338 Rolling the viewport causes unpredictable behavior when orbiting using turntable method. (blender.org)