Consider an interaction of the type (on a lattice) $$H=\sum_{\langle i,j\rangle}\left\{\left[\alpha c_{i\uparrow}^\dagger c_{j\uparrow}+h.c.\right]+\left[\beta c_{i\downarrow}^\dagger c_{j\downarrow}+h.c.\right]+\left(\gamma c_{i\uparrow}^\dagger c_{j\downarrow}+h.c.\right)+\left(\delta c_{i\downarrow}^\dagger c_{j\uparrow}+h.c.\right)\right\}$$ Here, $c_{i\sigma}^\dagger$ is the electron creation operator at the lattice site $i$ in spin state $\sigma$. The symbols $\alpha,\beta,\gamma$ and $\delta$ are four complex coeffeciients.
The first term in each of the two first brackets is spin-violating interaction because it destroys an electron at the site in the spin-down state but creates an electron at another site in the spin-up state. However, adding the hermitian conjugate term makes both $(...)$ interactions spin-conserving when considered as a whole. Thus it seems like, spin conservation is automatic by the hermiticity requirement of the Hamiltonian. My question is if we consider two-body interactions, does the requirement of hermeticity automatically ensure spin-conserving interaction? I hope that I have rightly interpreted the terms.