Can we find a pair of symmetric absolutely continuous measures on a measurable space?

Question

Let $(X,\mathcal{B},\mu)$ and $(X,\mathcal{B},\nu)$ be measure spaces.

Can we find a pair of measures $\nu$ and $\mu$ on $(X,\mathcal{B})$ such that $ \nu << \mu \implies \mu << \nu$, where $ \nu << \mu$ denotes $\nu(E)=0$ whenever $\mu(E)=0$ for any $E \in \mathcal{B}$ i.e. $\nu$ is absolutely continuous w.r.t. $\mu$ ?