Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $K$ and $K^{\text{'}}$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}\left(K\right)=g^{-1}\left(K^{\text{'}}\right)$. We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A{\bowtie }^{f,g}(K,K^{\text{'}})$ of $A$ with $(B,C)$ along $(K,K^{\text{'}})$ with respect to $(f,g)$.