Does a "commutative" asymmetric system exist?
RSA with random exponents would fulfill your key-swapping requirement. I.e. RSA where you generate one exponent randomly and then compute the other from it normally, with neither exponent made public. The operation to encrypt and decrypt is the same (modular exponentiation).

The security of this scheme, or some uses of it is considered in the (currently unanswered) question: Could this &quot;symmetric RSA&quot; scheme provide key compromise resistant communications?

Since you say you do not need it to be extremely secure, it should fit your bill regardless.