Perfect pairing co-weight lattice and root lattice

Question

Let $\Phi$ be a root system and let $\Lambda_R$ and $\Lambda_W$ denote root lattice and weight lattice. I know that there is a perfect pairing $\Lambda_W \times \Lambda_R^\vee \to \mathbb{Z}$, where $\Lambda_R^\vee$ is the co-root lattice. Is there also a perfect pairing $\Lambda_W^\vee \times \Lambda_R \to \mathbb{Z}$? Moreover, is the following string of inclusions correct (regarded as integer lattices in $\mathbb{R}^n$): $\Lambda_R \subseteq \Lambda_R^\vee \subseteq \Lambda_W \subseteq \Lambda_W^\vee$ ?