The Torah forbids wool and linen mixtures . The translation given below in Leviticus 19 of the word Shatnez is not accurate. In Deuteronomy, Shatnez is clearly defined as wool and linen.
The rabbis of the oral law looked at the verse 12 of Devarim
22, which commands the mitzvah of Tzitzit.
They somehow managed to confound this with the previous verse, and
arrived at the conclusion that whereas
Shatnez is an outright prohibition, in
the case of Tzitzit it is permitted,
namely the techelet thread should be made of wool, with the rest of linen. From here they also projected this synthesis
of the forbidden and the permitted onto the garments of the Kohanim in the
Leviticus Chapter 19 וַיִּקְרָא
אֶֽת־חֻקֹּתַי֮ תִּשְׁמֹ֒רוּ֒ בְּהֶמְתְּךָ֙ לֹא־תַרְבִּ֣יעַ כִּלְאַ֔יִם שָׂדְךָ֖ לֹא־תִזְרַ֣ע כִּלְאָ֑יִם וּבֶ֤גֶד כִּלְאַ֙יִם֙ שַֽׁעַטְנֵ֔ז לֹ֥א יַעֲלֶ֖ה עָלֶֽיךָ׃
19 Ye shall keep My statutes. Thou shalt not let thy cattle gender with a diverse kind; thou shalt not sow thy field with two kinds of seed; neither shall there come upon thee a garment of two kinds of stuff mingled together.
Deuteronomy Chapter 22 דְּבָרִים
לֹ֤א תִלְבַּשׁ֙ שַֽׁעַטְנֵ֔ז צֶ֥מֶר וּפִשְׁתִּ֖ים יַחְדָּֽו׃ {ס}
11. You shall not wear cloth combining wool and linen.
גְּדִלִ֖ים תַּעֲשֶׂה־לָּ֑ךְ עַל־אַרְבַּ֛ע כַּנְפ֥וֹת כְּסוּתְךָ֖ אֲשֶׁ֥ר תְּכַסֶּה־בָּֽהּ׃ {ס}
12. You shall make tassels on the four corners of the
garment with which you cover yourself.
Yechezkel – Ezekiel, reiterates the laws pertaining to the Priestly
garments, as if he prophetically foresees the abuse of this Law by the
Pharisees. Interestingly, he states that
it is the Priests of Tzadok who will maintain the
Ezekiel
44
17 וְהָיָ֗ה בְּבוֹאָם֙ אֶֽל־שַׁעֲרֵי֙
הֶחָצֵ֣ר הַפְּנִימִ֔ית בִּגְדֵ֥י פִשְׁתִּ֖ים יִלְבָּ֑שׁוּ וְלֹֽא־יַעֲלֶ֤ה
עֲלֵיהֶם֙ צֶ֔מֶר בְּשָׁרְתָ֗ם בְּשַׁעֲרֵ֛י הֶחָצֵ֥ר הַפְּנִימִ֖ית וָבָֽיְתָה׃
And when they enter the gates of the inner court, they shall
wear linen vestments: they shall have nothing woolen upon them when they
minister inside the gates of the inner court.
18פַּאֲרֵ֤י
פִשְׁתִּים֙ יִהְי֣וּ עַל־רֹאשָׁ֔ם וּמִכְנְסֵ֣י פִשְׁתִּ֔ים יִֽהְי֖וּ
עַל־מׇתְנֵיהֶ֑ם לֹ֥א יַחְגְּר֖וּ בַּיָּֽזַע׃
They shall have linen turbans on their heads and linen
breeches on their loins; they shall not gird themselves with anything that
causes sweat.
19 וּ֠בְצֵאתָ֠ם
אֶל־הֶחָצֵ֨ר הַחִיצוֹנָ֜ה אֶל־הֶחָצֵ֣ר הַחִיצוֹנָה֮ אֶל־הָעָם֒ יִפְשְׁט֣וּ
אֶת־בִּגְדֵיהֶ֗ם אֲשֶׁר־הֵ֙מָּה֙ מְשָׁרְתִ֣ם בָּ֔ם וְהִנִּ֥יחוּ אוֹתָ֖ם
בְּלִֽשְׁכֹ֣ת הַקֹּ֑דֶשׁ וְלָֽבְשׁוּ֙ בְּגָדִ֣ים אֲחֵרִ֔ים וְלֹא־יְקַדְּשׁ֥וּ
אֶת־הָעָ֖ם בְּבִגְדֵיהֶֽם׃
When they go out to the outer court—the outer court where
the people are—they shall remove the vestments in which they minister and shall
deposit them in the sacred chambers; they shall put on other garments, lest
they make the people consecrated by [contact with] their vestments.
In contrast to the claim of the Pharisees, no wool is permitted on/in any of the priestly garments. The rabbinic commentators try all kinds of pretzel logic to weave their way out of this – some say that it is only on Yom Kippur, and no shatnez was worn on that day (which is contrary to their own sources); Others say it was not Yom Kippur, and that these verses do not apply to the High Priest (although all priests would wear tekhelet according to the rabbis this would be wool woven with linen garments).
The absurdity of the zig zagging by the Pharisees is indicative that this matter is an Achilles heel for the oral law.
Rambam also has some difficulty in trying to solve this problem. Despite his protestations in his legal Mishneh Tora, that the priests' garments and also tzitzit were made of shatnez, in his Guide for the Perplexed he makes an interesting insinuation – pointing out that the ancient priests of Idolatry would wear shatnez tunics, and this is the reason why it is forbidden!