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INSIGNIA, AWARDS, RANK
To reply in the broad sense, understand the difference between "insignia" and "awards".
When I was with the 82d Airborne Division, I wore that unit's patch on the left shoulder of both the "Fatigue" and the "Army Green" uniforms.
I wore the branch insignia of Field Artillery (crossed cannons) and my 2d Lieutenant rank on both uniforms. On the "Fatigue" these were sown on the lapels (one on the right and one on the left). On the Army Green uniform the crossed cannons were pinned on both lapels and the 2d Lieutenant bars were pinned on both shoulders.
The airborne wings were an award so I wore them on both uniforms, over the left breast pocket. My award of the Ranger Tab is just as the name implies a shoulder tab. It is sewn on the left shoulder of both uniforms, just above the unit shoulder patch.
Upon departure from the 82d Airborne, I removed all the 82d Airborne shoulder patches from my "Fatigues" and Army Green uniforms. When I was assigned to a new unit, I would head for local tailor shop and have them replaced them with the patches of the new unit. An exception was when stationed at Headquarters, Department of the Army (Pentagon), there is no shoulder patch for that assignment.
When you serve with a unit in combat, you are entitled to wear the shoulder patch of that unit on the right shoulder of your uniforms forever after. I am entitled to wear either the MAAG Vietnam or MACV shoulder patch. (MAAG is Military Assistance Advisory Group and MACV is Military Assistance Command, Vietnam). These were the only units I have served with in a combat zone. If I had later been with say, the 82d Airborne in combat, (Dominican Republic, Grenada, Panama, Vietnam, Iraq, Afghanistan) I would have been entitled to wear the 82d Airborne patch on my right shoulder.
"Branch" insignia. When assigned to field commands, I would continue to wear my Field Artillery insignia. However at the Pentagon I wore the General Staff insignia, and at Sixth Army I wore the Inspector General insignia.
I am entitled to wear, forever and ever, my airborne wings (Parachutist badge) and Ranger Tab because they awarded to me as an individual. The same is true of my Combat Infantryman's Badge (CIB), my marksmanship badge, and my General Staff Identification Badge (for having been on the Army General Staff - Pentagon).
And of course "forever and ever" applies to my medals including the Bronze Star, the Meritorious Service Medal, my foreign award of the Vietnamese Cross of Gallantry with Bronze Star, the National Defense Service Medal, and various other campaign medals. These are my medals.
Additionally, if units you served with in combat were awarded unit citations, you are entitled to wear those. They are worn over the breast pocket of the Army Green Uniform. See the three bordered ribbons.
And last but not least, the unheralded Service Stripe. For the Army, this is a yellow bar (officers), stripe (enlisted) that is sewn on the lower right sleeve of the Army Green Uniform. One is awarded/authorized for each six months of overseas combat duty. I wear one of these. Why not two? Well my tour in Vietnam began in March 1964 and ended in February of 1965, 11 months, not 12. Illustration shows four stripes on the officer uniform indicating two full years of (overseas) combat service. God help us, let this award be one that is exclusively overseas.
I retired as a Lieutenant Colonel. The days of battlefield commissions are over, hence to get to Lieutenant Colonel, I followed the path of promotions from 2d Lieutenant (gold bar), 1st Lieutenant (silver bar), Captain (two silver bars), Major (gold oak leaf), and finally Lieutenant Colonel (silver oak leaf). Had I been promoted to Colonel, I would have worn the Silver Eagles. Then one star (Brigadier General), two stars (Major General), three stars (Lieutenant General) and four stars (General).