Is there a format preserving cryptographically secure hash?
Cryptographically secure hashes usually work on bitstrings of arbitrary length and output a fixed length bitstring. The secure part is being collision resistant and preimage resistant, so that you have a practical oneway function, and those are the properties you want for "scrambling".

As fgrieu psted in the comments, one easy way to do this is to utilize an encryption scheme, and use the plaintext for the key as well. That means, in order to decrypt the ciphertext you would need the plaintext. Formally:

$$F(p):= E_p(p)$$
with $E_k(p)$ denoting an encryption method with key $p$. If you want to utilize format preserving encryption, you might have to make adaptions how the plaintext actually is used for the "key input".

One more remark: Usually the key should be generated uniform randomly, and depending on your format, this might not be true. Therefore, if you use a normal hash function on the input $p$, then you should get a uniformly distributed hash value, and if you use this as key, you should be fine. Theoretic remark: In the random oracle model this should work just fine. So, we can adapt this to:

$$F(p):= \hat{E}_{H(p)}(p)$$

with a format preserving encryption function $\hat{E}_k(p)$ and a cryptographically secure hash function $H(p)$, where the key length $|k|$ is equal or less than the length of the output of the hash function $|H(p)|$.