To help you with the terminology, the names for the two operations are “signing” and “verifying”. That’s it.
What can you do with…
public keyprivate keyEncryption:encryptdecryptSignature:verifysign“Signing” is not at all the same as “encrypting” with the keys swapped. It is a separate specific sequence of mathematical operations you perform to combine two numbers (the private key and the message) to produce a third - the signature. Signing is not called “hashing”. A hash may be involved as part of the signature process, but it is not strictly necessary. It makes the “message” number smaller, but the algorithm can sign the full message without hashing it first, will just require computation for longer time. “Hash-verifying” isn’t a thing in this context, you made that name up, just use “verify”.
@dohpaz42 is mad because you messed up your terminology originally, and thought you were trying to say that you “encrypt” a message with the private key, which is totally backwards and wrong. He didn’t know that in your mind you thought you were talking about “signing” the message. Because honestly no one could have known that.