I too am sitting at my computer, but I can't see a cactus mug.
However, my reference(*) does depict the patch you describe...shield-like, cactus in upper right field, "rainbow" midway, with a tree in lower left field, and their motto ON GUARD on a scroll below the shield.
My reference includes your dad's unit, the 168th Infantry Regiment, along with the remarkable history of the parent, 34th Infantry Division.
His was one of three IRs in the 34th ID (133rd, 135th, 168th), plus other elements.
The 168th was a national guard unit from Iowa, taken into federal service 10 Feb 41 at Des Moines, Iowa.
The division trained at Camp Claiborne, LA; staged at Fort Dix, NJ.; 168th IR/34th ID departed NY P/E 30 Apr 42, aboard the CATHAY, as part of an eight ship convoy; arrived Ireland 13 May 42; arrived Scotland 23 Aug 42; assaulted Algiers 8 Nov 42; landed Italy 21 Sep 43; 133rd, 134th and 168th IRs/34th ID departed Naples, Italy, 10 Oct 45, aboard the MONTICELLO; arrived at Hampton Roads, VA P/E 3 Nov 45; inactivated at Camp Patrick, VA same day.
The division participated in these campaigns:
Algeria-French Morocco8 Nov 42 /11 Nov 42 Tunisia17 Nov 42 /13 May 43 Anzio22 Jan 44 /24 May 44 Rome-Arno22 Jan 44 / 9 Sep 44 North Apennines10 Sep 44 / 4 Apr 45 Po Valley5 Apr 45 /8 May45
Location at war's end, 2 May 45, on the Ticino River, northwest of Milan.
When the war ended in the Pacific, Aug 45, the division was at Iseo, Italy, where it was being trained for the anticipated invasion of Japan, which was scheduled for Oct 45.
Killed in Action2,866 Wounded in Action11,545 Died from Wounds484 That is over 100% casualties.
Their's was a long, hard campaign, with about as many days on the line as any other unit I know of.
(*) My reference:
WORLD WAR II ORDER of BATTLE, Shelby L. Stanton, author, Galahad Books, NYNY, publisher.
This is an excellent reference, which relates an incident when the 168th IR and 1st Armored Division became isolated on Djebel-Lessouda on 14 Feb 43, and then had to infiltrate through the German line during the night in order to get back to their own lines.
I have something in common with your father in that I also participated in the North Apennines and Po Valley campaigns. I was a chief-of-section in a 155mm howitzer unit, 985th Field Artillery Battalion, II Corps, Fifth Army.
I turned 20 at sea on the US GENERAL RICHARDSON enroute to Italy, and 21 in Mussolini's swimming pool in Rome.
I hope that this informaton is what you were after.