Difference between symmetric and asymmetric hash function? The Linux kernel supports symmetric and asymmetric hash functions. E.g. sha1, sha256, ...

See tcrypt.c and search for test_hash_speed and test_ahash_speed.

I know what the difference is between symmetric and asymmetric ciphers, where the encryption and decryption key is not the same.

Question

What is the difference between symmetric and asymmetric hash functions?
It is probably not the case of your example, but in some sense "asymmetric hash functions" do exists: they are called trapdoor hash functions (or also chameleon hash functions). 

Very briefly, they are collision resistant only if you don't know their trapdoor secret key.
Such functions take 2 arguments (instead of the usual one), and the second argument is used as randomness. 

So, without the trapdoor secret key it is infeasible to find 2 "messages" $m_1,m_2$ and 2 randomness values $r_1,r_2$ such that $H(m_1,r_1)=H(m_2,r_2)$; while, with the trapdoor key, you can efficiently compute (for any given triple $m_1,m_2,r_1$) a value $r_2$ such that $H(m_1,r_1)=H(m_2,r_2)$.

Hope this is clear enough. If you want, you can also see this  article by Shamir and Tauman or this one by Krawczyk and T Rabin