What the five sounds mean
Sa is sato, or sugar. Shi is shio, or salt. Su is su, or vinegar. Se is seuyu, an old spelling of shoyu. So is miso.
Sugar before salt is about size
A sugar molecule is about six times heavier than salt. Big molecules move into food slowly. Salt moves in fast and pulls water out. The food tightens, and sugar struggles to get in.
So sugar goes first and gets a head start. Salt follows a few minutes later.
The last three protect aroma
Vinegar, soy sauce, and miso all carry smells. Long heat drives those smells away. Add them later in the pot. Their exact order matters less.
When you can ignore the order
Many sauces are mixed before cooking. Gyudon and shogayaki cook for only minutes. The rule matters most in long simmers like nikujaga.
Try two small pots of nikujaga. Change only the order. Taste them side by side.