How to find current vertical axis

is there a way to find the current vertical axis of an object?

You can see the picture above. After rotation the Y axis is more like X and X is more like Y. But I still need a vertical rotation and in this case around X, but after many different rotation the axis change a lot and do not know which one is vertical - Y, X or Z. Can I always rotate around the current vertical axis?

transform.up does not work correct in my case.
By different rotations of the cube it loses the original place of the axis. For example the green axis (Y) may be changed to the place where was initially the X axis, as the picture above shows. In one moment I do not know where everyone axis is placed, but i want to rotate the cube from left to right as direction or the rotation must be Vector3.up, but in such a case it rotates again around the original Y axis and the cube rotates different. Is there a way to understand in every moment what is the orientation of every axis

On the picture the X axis now points up and then rotating to right or left must be around X axis.