Why is Symmetric encryption classed as Private Key?
Here's an easy way (in my opinion) of keeping it straight: Think of encryption as creating a "secure channel."

Symmetric:

Secret Key K &lt;-----------------------&gt; Secret Key K


Each side has the same shared key. Since they have a shared secret key, messages flow freely between them - they can both encrypt messages to send to one another. The picture is symmetric... the two sides match.

Asymmetric:

Public Key   -----------------------&gt; Private Key


One side has the private key, the other the public key. The public key holder can encrypt messages such that only the private key holder can read them, but the private key holder cannot use his private key to send secure messages to the public key holder (unless they negotiate/exchange some other key(s)). The picture is not symmetric anymore.