Tweeted and posted yesterday: Let me just say that a breach is where you have broken through the wall and a breech is where your ass is. You may be interested in more of the why and the wherefore.
Breach encompasses a sense of breaking. The noun can mean a broken condition, a break in a wall from battering ("Once more unto the breach" in Henry V), a violation of a law, or a break in previously friendly relations. The verb means to break or violate.
Breech originally meant short pants covering the hips and thighs, often as breeches or britches, later broadening the encompass pants in general. Breech also means buttocks, the part of the body covered by the breeches, the place at which the body forks into the legs. In childbirth, a breech delivery is one in which the fetus emerges buttocks/feet first instead of head first. In firearms, the breech is the part at the rear of the barrel, and a breech loader is a firearm in which the ammunition is inserted at the breech. A breeches buoy is a suspended canvas seat for transfer or rescue in water.
Broach, like breach, is associated with breaking. It can be an instrument for piecing, a spit, or a tool for tapping a cask. The verb means to break open, such as literally to open a cask of beer or figuratively to open a subject for discussion.
Broach, confusingly, since English is not necessarily an orderly language, is an alternative spelling for brooch, an ornament fastened by a pin or clasp.
It's your language, and it's your responsibility to make sure that what you say or write is what you mean.