Keyed reversible hash function? Is there such thing as a keyed reversible hash function? This would be a hash function where computing H(x) is easy and requires no key, but computing x from H(x) is only feasible with the key.

Is this absolutely synonymous with digital signatures? Or is this something else and does such a thing exist? It feels like the inverse of public key cryptography: encryption is public, decryption (reversal of algorithm) is private.
I think what you might be getting at is providing a signature that can only be verified by a particular party (i.e. who has access to the key).  Encrypting any normal hash value with the public key of the intended recipient would effectively do that.  The recipient could decrypt the hash with their private key and then re-hash the message to verify it.  Others would be left unsure of what the additional data represented.

However, if you're thinking of "hashing" the full message in a way that only the intended recipient reverse it, that is pretty much what defines encryption. And, as mentioned in the comments, public key cryptography handles exactly this use-case.

Either way, public-key cryptography seems to be what you're looking for, it's just a matter of how you want to deploy it.