In regards to Safety, Scoring, Course, Penalties, and advertising displays, yes.

They can make you add Safety equipment (I have never seen a Race subtract safety equipment.).

They can score versions of a boat either together or separately. For example Spin/nonspin F18's are all F18's, allow wings, allow carbon masts, or score all foiling boats the same. They can adjust ratings. They could say, in less than 3kts of wind, all boats will be scored as a single one design class called Iwanttogohome.

They can add a reach course to a class that precludes it or preclude a reach from a class that requires it.

They can say inadvertent contact is still contact and penalties apply while the class rules say otherwise.

They can make you take everything except your sail number and class insignia off your boat or add a giant ad sticker to your boat.

The sailing instructions can make a big difference.