We consider orientation-reversing Lozi maps $L_{a,b}$ in a parameter region $\mathfrak{R}$ where there are no homoclinic points for the fixed point $X$ in the first quadrant, and the period-two cycle ${P,P’}$ is attracting. We first analyze the boundary of that parameter region: we show that all homoclinic points for $X$ on that boundary are tangential or there is a segment of homoclinic points, and we classify them. Moreover, we study the topological entropy $h_{top}$ of $L_{a,b}$ when $(a,b)\in\mathfrak{R}$. Consider the set $\ell$ of accumulation points of the unstable manifold of $X$. Misiurewicz and Štimac have recently proven that $h_{top}(L_{a,b})=0$ in a certain open subset of $\mathfrak{R}$; in that case, $\ell={P,P’}$. We extend this result by showing that the $L_{a,b}$, restricted to the complement of $\ell$ in the plane, has zero entropy. Finally, we discuss that $\ell={P,P’}$ does not hold in general for all parameters in $\mathfrak{R}$.