There is an interesting Halacha regarding Shaatnez that is not very well known. The prohibition of wearing wool and linen is lifted when there is a Mitzva involved.
We apply the principle that עשה דוחה לא תעשה, that the positive commandment overrides the negative commandment. The Rambam in Mishna Torah, in הלכות כלי המקדש, discusses the various garments worn by the Kohanim. In addition to special pants, robe, and hat, there was also a special belt worn, אבנט.
This אבנט was made of both wool and linen and contained Shaatnez. It was permitted for the Kohein to wear this belt ONLY when he was performing his priestly duties. He had to remember to remove the belt as soon as he finished his Avoda. If he continued to wear the אבנט, when not on duty, he would be a Shaatnez violator.
This was not the case with ציצית, as it is a Mitzva to wear all day. An interesting Shaatnez law.