How are a private key and a seed phrase related? Is it a 1-to-1 relationship? If so, does that mean they are interchangeable? Can I then store just one of them and derive the other when needed?
It's the same relationship between the branches of a tree and the seed you put in the earth that grows into that tree.
We have a seed that can be used to derive child branches and child keys which makes the relationship is one directional (you can always derive the child key by having the seed but you can't compute the seed from the child key).
Take a look at the following picture from BIP32: