The Torah prohibits the mingling of species. Specifically, we are commanded against inter-breeding both animals and plants of different species, plowing a field with dissimilar animals (an ox and a donkey), and wearing a garment of linen and wool (shaatnez). Rabbi Zvi Cohen presents the issue of shaatnez in his book Tzitzit v’Tekhelet.
The first time the prohibition of shaatnez is mentioned is in Leviticus (19:19):
You shall keep my statutes. You shall not let your cattle gender with a diverse kind; you shall not sow your field with mingled seed; neither shall a mingled garment – shaatnez – of linen and wool come upon you.
It is repeated in Deuteronomy (22:11-12) with an interesting juxtaposition:
You shall not wear shaatnez, wool and linen together. You shall make fringes on the four corners of the garment with which you cover yourself.
The adjacency of the negative prohibition against shaatnez and the positive commandment of tzitzit is mentioned in the Talmud (Yevamot 4a, Nazir 41b-41a). Rabbi Elazar cites this case as an example of one of the rules of interpretation, namely: A positive commandment overrides a negative commandment.
In the case of shaatnez, the prohibition against wearing wool and linen together is overridden by the commandment of wearing tzitzit. This specifically applies to the case where the garment is made from linen, and the tzitzit on it are entwined with tekhelet, which is woolen.
However, this exception IS NOT to be put into practical use today. According to the Rama on the Shulchan Aruch (Orach Chaim 9:2) our practice is not to use linen at all in tzitzit.
It is worthwhile to note, that there exists another exception to the shaatnez prohibition. The garments worn by the High Priest were made from fine-spun linen and tekhelet. Much can be said regarding the similarity in status between tzitzit with tekhelet and these holy garments.