Why do ורחץ and רחצה have different names, and what do their names tell us about them?
1. The "vav" is so that all words have 2 syllables.
2. Also to stress that we wash after kadesh, even those that (back in the day) would wash year round before kidush. Alternatively, even if one washed beforehand, they wash over.
3. According to kabala, kadesh is chochma and urchatz is bina. The 2 are connected, hence the vav.
4. Rochtza vs rachatz is explained in pri etz chaim according to kabala.
Being poetic would warrant calling it rachatz. It's hard to explain but basically: by combining 2 together we have: urchatz, yachatz, rachatz. Then we have matza which will match with nirtza. Then we have korech, (shulchan) orech, berach. And nirtza which matched matza.