"And the Choshen (the High-Priest's breastplate) may not be untied from the Ephod (apron, one of the eight high-priestly garments), as Hashem commanded Moshe"

(Shmot 39:21)

One of the reasons for this Mitzvah is that Hashem wanted ... everything [in the Temple] to be perfect and set in its proper place... that their beauty should be unmarred. The point of this particular command is to ensure that the Choshen would not move and sway but rather would stand firmly in its place. This is how I shall understand this Mitzvah until I hear a better explanation.

(Sefer Hachinuch, Mitzvah #100 -- Moreh Nevuchim [3:45] suggests a different understanding of the Mitzvah which the Chinuch apparently found to be less appealing than his own)

One who appoints a justice who is not worthy [i.e., who is not well versed in Jewish law], is compared to one who plants an Asheirah (a tree of idol-worship).

(Sanhedrin 7b)

A judge who judges untruthfully causes the Divine Presence to be removed from Israel.

(ibid.)

Hashem, as King of the entire world, holds the responsibility to administer justice in His world. The Jewish judges are His emissaries, meting out justice based on the guidelines He set forth for them in His Torah. Judges are referred to as "Elokim" (Shmot 21:6 etc.) for, as Ibn Ezra explains, they are His emissaries, bringing about justice in the world, and an emissary can be referred to by the name of the one whom he is representing (see Sanhedrin 38b, "his name is like that of his master"- see also Ramban Shmot 21:6). When they announce a decision, it is as if they are declaring, "Hashem says to do such and such in this situation."

When proper justice, as dictated by the Torah, is practiced, it brings peace to the world. It is evident to all that the world has a caring king, who is "preserving it with justice." For this reason, "He who judges in a true and proper manner, causes the Divine Presence to rest among Israel" (Sanhedrin 7a). Inversely, when a judge misrepresents his master and dispatcher, he causes a desecration of the Holy Name. People are led to believe that Hashem cannot administer true justice in the world. This is what the verse means by saying, "Do not give anyone preferential treatment when judging...do not fear any person while judging, because Justice is the responsibility of Hashem" (Devorim 1:17). A judge is administering the responsibility of the Creator, therefore injustice that he causes will ultimately be attributed to Him. Thus, appointing an unjust judge can be compared to planting an Asheirah, for such a judge causes people to respect ideals other than Hashem's divine edicts, and to trust in them as coming from Hashem. This is why Rashi says (Shmot 21:1), "One who brings a quarrel before non-Jewish courts profanes the Holy name and shows honor for idol worshippers, praising their idols... as if to bear witness to the superiority of their Gods."

The command that the Choshen may not be untied from the Ephod reminds our judges that the justice which they are appointed to administer is representative of Hashem's Own word. Inevitably, if they administer Hashem's laws faithfully they will bring the world to respect Him as Creator of the universe. On the other hand, if they misrepresent His laws they are misrepresenting Hashem, which is tantamount to idol worship!

Perhaps the permanence of the bond is also hinting that even when he is not in court, if a judge acts in an improper manner he will cause disrespect for his Creator. His actions will be viewed as a reflection of Hashem's will.

(I later found that Yalkut Yitzchak, Mitzvah #100, cites in brief a similar explanation from Ta'amai Hamitzvot [of the Rikanati].)