I am learning about partially homomorphic cryptography, and was interested to see if there was a system such that one operation was homomorphic, but its inverse was not.
For example, if I have two messages $m_1$ and $m_2$, with an ecryption scheme $e(x)$, is there a cryptosystem such that we can compute $e(m_1 + m_2)$, but it would be infeasible to compute $e(m_1 - m_2)$? Or a cryptosystem that can compute $e(m_1 m_2)$ but not $e(m_1 m_2^{-1})$?
I looked into homomorphic addition systems such as Pallier, which is feasible to compute the additive inverse. I also looked into homomorphic multiplication systems such as RSA and El Gamal, but it seems like its possible to compute the multiplicative inverse as well. Is it always the case that if one operation is homomorphic, its inverse is also homomorphic?