This problem comes from R.Hartley & A.Zisserman Multiple View Geometry in Computer Vision at page 78, the Result 3.6 says
Any particular translation and rotation is equivalent to a rotation about a screw axis together with a translation along the screw axis. The screw axis is parallel to the rotation axis.
To my understanding, order of rotation and translation will make a difference on Euclidean translation. because
$$Rx+t \neq R(x+t)$$
Now decompose translation into $t_{\bot}$ and $t_{||}$, and do screw decomposition as follows:
So whether or not it doesn't matter when exchange the order of translating $t_{||}$ along $\mathbf a$ and rotating about $\mathbf a$?