In a local discussion someone said the private key (of a private/public key pair) would also be a symmetric key.
Well, not quite. If we're talking about public key encryption (there are other flavors of public key cryptography), what is correct is that "what has been encrypted with the public key can be decrypted with the private key".
What this means is that to encrypt, you need the "public key". That is, unless the private key had enough information for you to deduce the public key, you couldn't actually encrypt, and so it wouldn't work as a symmetric key.
Now, practically speaking, in most public key cryptosystems, the private key does have enough information to deduce the public key. However there is at least one example were it doesn't; large exponent RSA; where both exponents $e$ and $d$ are chosen to be large (and you don't insert the CRT parameters into the private key). With this system, given $N$ and $d$ (which is all the private key has), you cannot efficiently recover $e$ (and hence cannot encrypt, even though you can decrypt).
Of course, a public/private key pair, when considered as a unit, will work as a key in a symmetric system. We do run into the practical issues that it is a hideously bad symmetric system (as any public key cryptosystem is slow and has huge ciphertext expansion, compared to any realistic symmetric cryptosystem), but it would work.
As a side note: if your document states:
I found places that document "what has been encrypted using one key of the private/public key pair can only be decrypted with the other key of the pair."
Then either the document had context that stated that it was specifically talking about RSA, or it was written by someone who doesn't know what they're talking about. Most public key cryptosystems cannot be described this way; they do not provide any way that allows talking about 'decryption' (recovering the original message) with the public key. RSA can be described as a 'trap door' permutation (if you ignore padding, which in practice you can't), and so could many multivariate signature schemes (although those are mildly obscure); hardly anything else can be.
