You shall not mate your animal into another species, you shall not plant your field with mixed seed; and a garment that is a mixture of combined fibers shall not come upon you. (Lev. 19:19)
There are three different types of forbidden mixtures dealt with here, which can be categorized into two classes. While there are certain instances where clothing may contain a combination of wool and linen, such as the garments worn by the priests in the Holy Temple and a garment bearing tzitzit, the crossbreeding of both animals and plants is never allowed.
This may be understood in light of the differences between the two. The prohibition against combining fibers is a seemingly illogical law. After all, even after they are woven together the fibers remain two separate entities; it is possible to completely unravel and reverse the combination. Crossbreeding, on the other hand, is a readily understandable law. G‑d created many different species, and commanded them to remain pure and true to their nature. By tampering with them, man is disturbing the order of Creation and causing a different, new, and confused entity to come into being. The product cannot be considered the result of either of its two antecedents, but rather a brand new entity. Regardless of when or where, this is something that can never be allowed.
Why, then, can fibers indeed not be combined, and why is an exception made for the priestly garments and tzitzit?
Rabbeinu Bachya explains the Kabbalistic secret of shatnez, the prohibition of combining wool and linen, as follows: One who wears such a combination jumbles the spiritual energies…
When one mixes two incompatible species, this causes friction, for he is jumbling the supernal energies and changing them from their proper state…. The two firstborn of the world [Cain and Abel] were such a mixture, one good and one bad. We are enjoined to follow only the spirit of holiness and to distance ourselves from that which is evil and impure. This is why such forbidden mixtures are forbidden, for they fuse the two opposite extremes, and their combinations do not work well…
Of the two firstborn, one [Cain] brought flax [linen] as his sacrifice, while the other [Abel] brought wool [sheep]. Therefore, we may not wear a mixture of wool and linen, for one who wears such a combination jumbles the spiritual energies…. Jews are forbidden to wear this combination of opposites in order to stay away from the spirit of impurity and in order to become sanctified with the spirit of holiness.
Avoiding any influence of that which is impure, implies that there is no problem with the mixture per se, but rather that by combining these two species the holy is being tainted by the impure. The problem is in the fact that they are being combined, even if there is no problem with either of the two components. Each one is a different species, and it is unnatural for the two to be mixed. In spiritual terms, this causes a merger of different energies, which results in one or the other being perverted from its proper mission. Cooperation is fine - losing oneself in another's identity is not…
Although all spiritual energies are supposed to coexist and overlap, that is only as long as neither lose their identity altogether.
Cooperation is fine - losing oneself in another's identity is not. This can be compared to the way a king desires his ministers to behave. He wants them to work so well together that their different portfolios integrate seamlessly. Nevertheless, each one has his/her own specific mission, which is distinct from that of every other one. If these separations are lost, the king will definitely not be pleased.
This is true, however, only when the ministers are functioning independently of the king. When they are in the presence of the king, they lose all sense of self, for they are totally nullified before the over-powering essence of their master. Here, any demonstration of self-assertion would be out of place and an affront to the king's majesty. As they bow before him, there are indeed no differences between the ministers, and even those that are total opposites become one. In the Holy Temple…the impurities inherent in the flax could not exist…
So it was in the Holy Temple. In the face of its holiness, the impurities inherent in the flax could not exist, and so its presence would not sully the purity of the wool. And, because of the intense degree of spirituality manifest while the priests served G‑d there, even diverse, contrary energies were peacefully joined.
In fact, this is the root of the difference of opinions between two early great Torah authorities as to when this exception applies. Rambam maintains that the priests could only wear the priestly garments in the Temple while actively involved in the priestly service. Ra'avad, however, disagrees, saying that they could wear them at all times, as long as they were on the Temple premises.
Ra'avad considers the prohibition against shatnez to be because of the evil of the flax. Since evil has no place in the Temple, there is no problem at all with shatnez in those holy confines, even when one is not involved in actual service.
Rambam disagrees, viewing the convergence of disparate energies as being the main issue, resulting in the prohibition against shatnez. Even while in the Temple, these energies are present and retain their differences. It is only while directly involved in the divine service that G‑d's overpowering Oneness is manifest, and all distinctions lose their meaning. It is only then, therefore, that the combination of wool and linen may be worn.
[from Likutei Sichot, vol. 29, pp. 122-9, vol. 36, pp. 155-7]