Difference between symmetric and asymmetric hash function?
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