I did not flip any signs

Yes you did! 😂

merely reversed the order in which the operations are written out

No, merely reversing the order gives -3-2+1 - you changed the signs on the 1 and 3.

If you read the right side from right to left, it

Starts with -3, which you changed to +3

it has the same meaning as the left side from left to right

when you don't change any of the signs it does 😂

Hell, the convention that the sign is on the left is also just a convention

Nope, it's a rule of Maths, Left Associativity.