What do you mean by secure encryption algorithms which are reversible? Encryption implies an operation in reverse order to exist, otherwise it is in essence, digesting.
It's always (almost) completely unnecessary to store encrypted passwords.
> What do you mean by secure encryption algorithms which are reversible?
AES-256 symmetric encryption (as one example) is designed to be reversible while a SHA-512 hash is not. What does reversible mean you ask? It means that plaintext can be made into ciphertext and then back to plaintext again. It's designed to be undone/reversed by the party that holds the key/password.
Reversible does not mean the same thing that symmetric does. Encryption, by definition, is reversible. If your algorithm is not symmetric, you will not be able to decrypt it with the same key as you used to encrypt it, but it would still be reversible. If it wasn't reversible, it would be data destruction (ie, Unix shred) or hashing (ie, SHA). "Reversible encryption" is a pleonasm.
It's always (almost) completely unnecessary to store encrypted passwords.