Thank you for your reply.Since my intial post I called the CSM at Fort Wainwright and talked with his assistant.(I don't know why I didn't think of this earlier).He said the same as you did - that the 159th was eleigible to wear the shoulder insignia (SI)of the 106th Infantry Division (ID).
I have two pictures of my father from the time he was with the 159th and neither show a SI on the uniform.Among his WWII possessions is 106th ID SI but he also had SIs from other IDs.So was the 106th SI just something he collected or did he wear it?My father past away in 1982; I wish I asked more questions before way back then.
I emailed the 106th from an address off their web page but never got a reply.
As to the 159th wearing the 106th ID SI in the ETO, the question is and as you put it - "would they even want to?"And this question is not meant to be specific to the attitudes and history of these two units but to attached units to a division and vise versa in general.The Army found fault with its own replacement system when it came to individual soldiers in a post war analysis. Does that extend to higher levels in the command structure as well?