the way i understand it is that signing/verifying messages is different from encrypting/decrypting messages.
(Corrected after re-reading your message)
I see them similar to what you wrote. Indeed, encrypting should involve a transformation that will allow through decryption to restore the original message. I don't know if this is what OP wants, hence I didn't try to answer to that part (although I know that BitCrypt does this).
And since OP also said he is interested about signing and verifying messages in a simple way, I tried to cover that part.