Is a transposition cipher categorized as asymmetric algorithm? Consider a simplest transposition cipher, such as one with block size 8 and encryption key 43725681, the decryption key will be 84215637. As the keys are different, may I say a transposition cipher is an asymmetric algorithm, except for some special values of the keys, such as 87654321?
No, it is not an asymmetrical algorithm, becayse the decryption key can be derived from the encrytion key.

Even in a symmetrical algorithm the encryption and decryption algorithms may be different, and the decryption algorithm might include firct calculating a decryption key from the encryption key. However, the same key can be used for both encryption and decryption in the sense that if you know that key you are able to both encrypt and decrypt messages.

In an asymmetrical algorithm, someone who only knows the public key can not decrypt messages. That is not the case for a transposition cipher.