Set normal orientation of origin I want to change the orientation of my object to a specified direction.

Say I have an axle of a train whose normal orientation is set off by 90° from the global orientation:



The green arrow (Y-axis) points to the back of the train (along the global X-axis). But I want it to point to the sides of the train, so all I want is to rotate the orientation by 90° without rotating the object itself.

In other words, I want the normal and global orientation of the axle to be the identical:



Is this possible and if so, how?

Background:
The game engine to use this model will move the bogies and the axles according to their orientations. After importing this train into the game the axles rotated 90° off.
Select the object in Object mode, then CtrlA > R (Apply rotation). This resets the rotation to 0° around all three axes (and thus aligns the local axes with the global axes) without rotating the object.