Interchangeable role of public/private key pair and encoding/decoding algorithms In designing a public-key cryptosystem, there are often two desirable properties:


Public/private key pair of each participant can be exchanged. Meaning that whenever a participant has generated a key pair $(a, b)$, it is free up to him to choose to public either $a$ or $b$ (but not both, of course)
Encoding/decoding algorithms $(E, D)$ are interchangeable. It is up to all participants to choose either $E$ or $D$ to be the encoding algorithm, then automatically, $D$ or respectively, $E$ will be the decoding algorithm. But the choice must be made prior to any communication.


Note that if $D$ and $E$ are identical, for e.g. $\mathrm{RSA}$, then the property 2 is vacuously true.

The question is: Assume that $E$ and $D$ are different, is there any known public-key cryptosystem that satisfies property 1 but violates property 2.

Informally, the cryptosystems that we are interested are the ones in which the key generation is in some sense symmetric, but the encoding and decoding algorithms are not.

EDIT: We only consider deterministic encryption scheme. To be meaningful, we assume $\mathrm{M} = \mathrm{C}$ and $\mathrm{PU} = \mathrm{PR}$ where $\mathrm{M}$ is the message space, $\mathrm{C}$ is the ciphertext space, $\mathrm{PU}$ and $\mathrm{PR}$ are public and private key space.
Any RSA variant that goes beyond textbook RSA (which is insecure) will probably not have symmetric encryption/decryption.

Examples are RSA + OAEP or RSA used to encrypt an AES key which is used to actually encrypt the message.