One relatively minor thing I've never understood was the designation of Left, Center, and Right Wings. It seems to presume that they army will always be in that configuration, no matter how or in what direction it maneuvers. I suppose there's no great harm if say the unit called Center ends up on the left, but it could be confusing. Nor does that system seem to provide any benefit over the simple 1st, 2nd, 3rd, etc.
Classically, seniority was indicated by the names. This goes down to where was easiest to control formations in line from, and was even operative in the Roman Army.
The right-wing was the senior wing, the left wing second and the centre wing third. By tradition:
Right = 1st
Left = 2nd
Centre = 3rd
It was generally understood that the 1st division would form the right of the corps, the 1st brigade the right of the division, and even the senior regiment the right of the brigade. These deployment were practiced and hence easier than other deployments. When something occurred to make the deployment happen in another manner, there was added confusion.
Eventually, numbering of Corps within the Federal army had nothing to do with assigned positions, and looking at the Army of the Ohio/Cumberland, the rules are clear:
Buell divided the army into three corps:
1st Corps - right wing, under the senior corps commander (McCook)
2nd Corps - left wing, under second (Crittenden)
3rd Corps - center, under junior (Gilbert)
When armies march it is traditionally the right at the front, centre following, and left in the rear. Buell's column at Perryville was marching north and follows the traditional scheme. The Perryville fight was essentially entirely maintained by 1st Corps.
When Rosecrans took over the whole army was made into 14th Corps. This is a paper thing - Corps commanders had an additional Commission appointing them to the numbered Corps signed by Lincoln. This effectively made permanent appointment to a Corps a promotion within the grade of MG, since Lincoln had make you a CC, and your army commander couldn't reassign you.
When the whole AoC was made 14th Corps, Rosecrans renamed them wings, in the traditional manner. They moved and fought in exactly that formation (see Stone's River). The wrinkle was that Thomas replaced Gilbert. Gilbert's brief tenure commanding 3rd Corps Army of the Ohio, explains why seniority looks a little off in the later AotC. Shortly thereafter, 14th Corps was divided in three (14th, 20th and 21st Corps), but if you look at Chickamauga etc. they still stick to their old formation; 1st Corps/ RW/20th Corps always on the right, 2nd Corps/ LW/ 21st Corps always on the left, and 3rd Corps/ CW/ 14th Corps always in the centre. Checking Kennesaw Mountain, they deploy as per tradition...
Why these traditions developed was just to give a regular procedure.