I'm working on the surface $X$ which is birational to the blowing-up of $\mathbb{P}^1\times \mathbb{P}^1$ at two points. When I consider its cohomology group $H^2(X,\mathbb{Z})$, I can use a basis as $(L_1,L_2,E_1,E_2)$ with $L_1,L_2$ are the pull-back of two generators of $H^2(\mathbb{P}^1\times \mathbb{P}^1,\mathbb{Z})$ and $E_1,E_2$ are two exceptional divisors. But when I represent a curve $H^2(X)$, it can not be symmetric (That means: for example a curve in $\mathbb{P}^1\times \mathbb{P}^1$ can be presented symmetrically as $aD_1+bD_2$ or $aD_2+bD_1$ by changing the basis $(D_1,D_2)$ into $(D_2,D_1)$ in $H^2(\mathbb{P}^1\times \mathbb{P}^1,\mathbb{Z})$. But (the proper transformation) curve in $X$ becomes $aL_1+bL_2-\alpha E_1-\beta E_2$, then it is no longer symmetric).
So I would like to ask if you know any others basis for $H^2(X,\mathbb{Z})$ such that we can have a symmetric representation for its curve?